Okay, I've tried to grasp this one by the book, but came up with nothing.
Does stuff affected by Muto magic always return to its former shape when the duration of the spell ends? For example, would a door affected by Pass the Unyielding Portal (MuHe 5) stay bent after the spell expires? Does the Limit of Essential Nature come into effect here, or is a bent door still a door (or, closer to my problem, a molten wall still a wall)?
Yes definitely...sometimes
I look at if some other non magical activity has effected it while muto-ed...
eg: A rock changed to clay and then shaped by a potter will not change back to a stone lump, but will be a stone pot.
A stone wall changed to liquid and allowed to flow down hill, will reform on the floor as if you had dumped a load of concrete and allowed it to harden.
That is my opinion also.
Basically the alteration that the Muto magic makes on the target is undone. However any other changes done on the target by other means, even means not normally possible without the altered properties of the muto magic, are not.
So, for example, I wouldn't say your 'could' use Muto magic to simply bend a door.
You could however make the door more pliable/malleable, bend it, and then drop your spell. The door would then harden back to it's original state and stay bent.
so then you could make chain mail more easily by making the steel softer until you had each ring shaped and fitted... then the steel is as hard as it was prior to shaping without the weakness that warping the steel normally causes?
Vortigern has a point. Muto "Transmutes", it doesn't Tansform (regardless of what spell titles may say). Muto changes the properties of a substance, or changes what substance an item is made from. This can lead to a change in shape, but not necissarilly so. The Transmutation wears off, gold turns back to lead and so forth. But whatever has affected the item does not change. So you can muto steel into liquid and pour it into a mold, and when the spell ends the steel becomes solid but is now in the shape of the mold.
With Rego, you can alter and change the shape of an item all day, so long as you do not change its substance. You can use Rego to force gold into the shape of an urn, and when the spell is over it retains that shape.
I think so. I was unaware that forming the steel into links weakened it. I am not a metalurgist. Mechanically I would give the mail no special advantage or bonus. I would just say that it makes the process easier and quicker, therefore less expensive and time consuming.
Wouldn't that also require Muto, at least as a requisite (since gold is normally solid)?
Not necessarily. In broad terms, imagine that rego gives you the abilities of a skilled arttisan at level 10 with that substance. If you have a bunch of logs, and nails you can build a stout house rego-ing the materials to place, for example. If a skilled artisan in his labor site could make the gold fit into the pot (heating it during part of the process, for example) you can achieve the same result using rego terram to affect the gold. The heating et al is implicit in the use of magic here 
Essential Nature would only come into effect if a particular shape was that thing's essential nature. I would say that a wooden door's essential nature is that of "being wood", not of "being a door". The difference is a temporary shape (here, one crafted by mundane hands) vs the thing's essence, how it grew naturally. The grain, the hardness, the tendency to burn and rot and float, those are all elements of the "essential nature" of wood. Being a door is just coincidental.
(Otherwise, anything could be changed into anything else, and then we claim that new thing is the "new essential nature"! EN doesn't change, that's what makes it "essential", part of the "essence" of the thing. EN tends to refer to a substance or a living entity more than a mere shape or final product. So, wood or steel has an EN, but I'm not sure a door or chainmail would, depending.)
Muto magic can basically do one of two things, both of which are "unnatural"*. One is change the shape, and that includes adding unnatural shapes (like wings or claws or whatever) to "normal" things. The other is change the substance itself, either giving it unnatural qualities or changing it into another substance.
(* This is an important distinction, because that's how Rego gets to do "natural" things to substances, even if there is art/craft involved. Thus, wood can become a staff or a violin or a puppet via Rego, but not apple wine or a linen shirt.)
If magic is used to change the shape, then when the magic wears off, the shape returns to "normal". If, however, magic is used to change the substance, then the substance merely returns to the original substance when the spell ends.
In both cases, the "shape" can be changed mundanely - by cutting off a part of it with a blade, or pressing in a part of it with a tool, or bending it into a new shape (if it bends). This might be easier with some substances, but in either case the shape has been changed mundanely and is not tied to the magic. Change a man to a dog, and break the dog's leg, and the man will later re-appear with a broken leg. Change a man into a rope, and tie the rope into a knot... oooh.... 
So, in your chainmail example, it could be assumed that if you change the metal to something that can be easily bent (say, lead), and then it changes back, it changes to exactly the substance it was - not weakened. (But nor could one harden a metal like copper or bronze more easily this way, to change it, work it, and then change it back to attain the hardness that would come from such working.)
In Xavi's above example of Rego, the change is something that could be achieved "naturally" (with the proper craft and tools) - if you feel that Muto should be a requisite for some tasks/final products, that's your call. (It's not in the core book, but discussed elsewhere - A&A, iirc??? Someone will confirm/correct.) But, as defined, Rego doesn't simply "move stuff around" - it changes it's current, natural state - if the current state is a log, then another possible natural state of that substance is lumber - and if that, then a chair or a house, or a violin, or whatever.
Muto changes the substance itself, for the duration of the magic, then the item returns. Mundane changes are incidental to the course of the magic.
Certainly not. You can shape solid steel as long as you apply enough force. And its been commonly used as long as history of using metals goes.
en.wikipedia.org/wiki/Cold_working
en.wikipedia.org/wiki/Cold_rolling
And gold is so soft that it can easily be worked on cold.
Which is why most jewelry and other uses of gold rarely uses pure gold, its just too soft to stay in shape.
en.wikipedia.org/wiki/Goldbeating#History
YEah, Rego can turn a tree with branches into a log with scrap to the cide and then into lumber but it can't reverse the process. Muto can turn your lumber back into the tree for the duraction..
Rego can work things but only only the natural progression. Muto is required for any unnatural change.
Exactly.
Is that clear(er?), Toa?
The AM model of magic is not always intuitive, and rarely as simple as one might at first assume. The more you think about it, the more those subtleties are important (esp if you played earlier editions, where the eventual occasional paradoxes and contradictions were not as well addressed. And yet, those versions were still leagues ahead of any other games, and won awards for same - and for good reason.)
I'm assuming that the standard 'turn a tree into a lumber' would necessarily be a rego spell right? (excepting a muto spell with some other intermediary event... i.e. turn a tree in to liquid and pour into 'lumber' molds). What exactly are the limits defining 'natural' and 'unnatural' change?
I would say a natural change is something that can be done by application of tools/force. The process of raw materials into finished goods (duplicating work of a crafter) is Rego. If it can not be done by a crafter or laborer, then it is not rego.
Also if it is not the same substance at the end, it is not rego. This is more my opinion since it is grey. If you have copper or and tin ore, I think it would be muto to make them into bronze, not rego. I am sure others might disagree. If you have sand and you want to end up with glass. That would be muto, not rego (maybe rego with Ig requisite but I think muto). If you have iron ore and want steel, that would be muto, not rego.
Thanks. That was my take as well. I brought the question up because your original example could be misleading. It starts with a Rego effect and then comments on how a hypothetical Muto effect would end, possibly implying that you could use Muto to accomplish the Rego effect. This, as you have confirmed, isn't strictly true - you can only do so by choosing a Muto effect that results in the same 'mundane' result when it ends.
With the Ars magic system there is usually more than one way to skin a cat - I'd think that you could use Muto (tho' it's not the best choice) to shape one tree into many boards, or even Perdo to accomplish what a saw/blade could to a tree. (or to a cat, tho' that was just a turn of phrase. But same diff.) 
Rego simulates "crafting" an item, regardless of how skilled or mundane that craft may be - chopping down a tree or creating a violin from that wood can both be "Rego", as well as other Techniques (tho' if "aging" wood is necessary, perhaps not so much with that latter example). Rego cannot do what Muto can if it's unnatural, and some final effects might require SG rulings.
Keep in mind that there are many ways to do things.
An example is flight: you can fly by creating wings to lift you (wings of the soaring wind), control the physical body to lift and move it (rise of the feathery body and several spells on the Ars Wiki), control an object (rete for a flying platform), perhaps transform into a bird (muto Corpus).
You can MuHe a tree into a board or piece of lumber but when the spell ended, it would return to being a tree. Basically when the muto spell ends, whatever was changed is no longer changed. so if you changed the shape, the shape reverts. If you changed the material, the material changes back. Any mundane manipulation of the changed object though is reflected after it changes back. Wood to water, the water becomes wood when spell ends but the wood is whereever the water was in shape the water was.
I fully get the 'many ways to skin a cat'. I just made an interjection to the original example because it didn't make reference to 'a different way of skinning the cat' nor a 'different, unnatural effect', when making the jump from Rego to Muto and I thought one (a newbie) could come away from that with the wrong impression.
To further expound:
Assuming that the magic effect of the spell is to directly turn a tree into boards, you must first decide if such a state for tree can be said to be 'natural'. If it is, then Rego. If not, then Muto. As such, if you want to create the same result (if not the same magic effect) you could come up with alternate methods such as the 'liquid wood' solution. I'd venture you could even add a Rego requisite in order to 'mold' the liquid magically (as opposed to using mundane molds).
I'm assuming that the inverse is true... Muto can't do what Rego can if it's natural. That, once you determine that the magical effect (again, independent of the result) is natural, you've eliminated Muto from the equation.
That's my understanding, although I'd be happy to entertain alternatives if I'm missing something here.
More than one way to Skin a Cat 
Over on the Berklist, we had declared that "Skinning a Cat" was considered a Major Focus, because we were able to think of how to accomplish it using just about every Technique & Form combination possible.
T - I assumed "you" did - it's "everyone" I was aiming at. 
Well, let's see if we can find examples/counter-examples.
If you had a section of tree, and wanted to create a wooden bowl, Rego could produce something as if it was carved from that bowl. Muto would simple change that entire thing to a bowl, including shrinking/expanding it if/as necessary. The difference in the end produce would, I think, be one of aesthetics - with Muto, what you would have created would be closely tied to Platonic Forms - you'd have a generically perfect wooden bowl, and anything artistic would be a stretch with Finesse. With Rego, it could be as artistic as you want, and that end of things would be tied to both one's Arts and also Finesse.
(Perdo could do the same, by removing the material. Probably no more artistic than Muto.)
The challenge for Muto comes when a complex/compound finished product is involved - say a table. With Rego, it would look like a hand-crafted table, the legs and top joined and pinned and reinforced, etc. With Muto, it would all have "flowed" from the one piece of wood, and look (at least upon close inspection) inexplicably magical in construction.
And if we move to something that requires an artist's touch (at one level or another), such as inlay work on that table, or a musical instrument, then Muto is definitely at a disadvantage - it could make the shape and appearance of a violin (from wood*) or silver flute (from a chunk of silver*), but not the individual pieces, nor certainly the quality of tone (or at least not as easily.)
(* Higher magnitudes of Muto could start with anything and change it to anything, changing a skinned cat into a grand piano or whatever, natch.)