That's one of the results, yes. Note that the magi in question still needs to gain Insight from each scroll - meaning that they need to be able to roll an 18+ on a Magic Theory roll in order to comprehend each source. (I don't recall if they need to study it for a season beforehand, and then learn the spell, though. Personally, I houserule that if it's already been translated into a Hermetic lab text, another magi can spend a single season, rather than a minimum of 2. But this may not actually be a houserule, but rather RAW.)
EDIT - OK, went and looked it up; yeah, in the RAW, you have to spend a season learning the Insight; you don't necessarily have to get the Insight, then learn the spell, though. So technically you could re-create all the spells from the Lab texts, and then gain the insight - this might be a bit quicker, as some of the spells may potentially use the same TeFo's in the lab, and you might be able to learn 2 in a single season.
This also strongly suggests that if you want to speed up Integration, you can Initiate into an existing Virtue, and then do your research. In theory, that would reduce the time required by half. The first part of Integration (ie, getting the virtue)simply means "you understand it well enough to do it yourself". The second part of Integration is "making sure EVERYONE can do it." However, if you've already got the virtue (either by simply being born with it, taught it, or Initiating into it), then you're already well on the way.
Nope. Partially-integrated virtues aren't a part of Hermetic Magic yet - at best, they're part of the personal art of the individual that has them. So while a given virtue may be well-known and taught (using the Apprentice rules) or Initiated (using the MC rules), it's not yet a part of the Hermetic canon yet.
Nope, this would simply be the 2nd part of Integration, rather than original research. If a magical source exists that you can copy/study, you don't need to do original research - you can continue to study your Source for Insight. So, in the Elemental example, the magi in question (once they got the Elementalist virtue) could simply continue to study his Elemental sources - likely the Hedge tradition itself - and perform a complete integration. This would take an additional amount of Insight (the same as it took to get to the virtue - 45, in this case). Once that was done, the magi could write the aforementioned tractus. Anyone who spent a season reading it would then get an XP bonus when studying the elements.
EDIT - you could do original research if you wanted to, though - the Breakthrough points from Integration and Original Research are identical in their results. As such, you can mix-and-match: if you didn't have a local Elementalist tradition to study for Integration insight, you could simply go into your lab and use the Original Research rules instead.
Non-hermetic (but still supernatural)? Sure. That's actually what Elementalism is, as "Elementalism" is actually the Hermetic, partially-integrated version of the similarly-named Supernatural virtue.
In terms of general virtues? ....I would personally say "yes, if it makes sense" - For example, "Immune to the Gift" is a general virtue that makes you immune to the Gift. An enterprising magi might try to figure out how to get that into a ReVi ward - essentially duplicating that part of Parma. However, that's an example of an explicitly supernatural effect which happens to be duplicated by a General virtue. Your explicit example (Learn from Mistakes)...I would personally say that such a virtue isn't specifically magical, and as such can't be integrated into Hermetic magic theory. If it could be shown that it's specifically interacting with magic (as opposed to simply being a general virtue which is currently applied to an arcane ability), then sure.
In terms of Minor/Major/Hermetic? I'd say it depends on the effect. The fact that something is a General virtue instead of a Supernatural one doesn't affect what kind of Integration it would be. Rather, it's "how far outside current hermetic thought is this new technique?" For example: in terms of game mechanics, the Gift itself is a 0-point supernatural ability...but Integrating the ability to grant the Gift is a Hermetic breakthrough.
EDIT - actually, in thinking about it a bit more: the Learn from Mistakes example, if integrated, wouldn't be Integrated into Lab activities - rather, it would be integrated as a Mentem "grant mental virtue" guideline. This might technically already be an implicit guideline: CrMe rituals allow for mental attributes to be +4 and +5, for example. However, that's integrating a magical ability, rather than the "learn from mistakes" ability into the process of Hermetic lab activities.
Example: Figurine magic (from HoH:TL), I believe, can grant general virtues. But again - that's an example of a magical ability to grant virtues, rather than integrating the virtue itself.