If a magi casts a rego spell to move something but another being or effect tries to resist how do you resolve what happens?
For example, a maga tries to make a staff fly to her hands, but 2 soldiers are holding on to it. Was is the level of force involved? Or a giant grabs one of your shield grogs and your magus attempts to pull them out of it's grip with a rego corpus effect? Who wins and is there any damage to the shield grog?
Roll some dice. 
To be a bit more helpful....Who gets the item the tug of war is being fought over would be a contested strength roll, you'd just have to determine the strength of the spell.
The shield grog is in a world of trouble. If the spell has enough strength to take the grog from a giant, we then have to ask does all the grog come with the spell, or does some of the grog remain with the giant, resulting in a sub-optimal outcome for our grog.
I'd be thinking do the opposed strength roll and then add the results for total damage.
For example - Lets say the opposed roll was 13 for the spell, 11 for the giant. 13 wins, spell takes grog from giant. 24 points of damage - soak (no armour) for the grog.
If there's an answer in a supplementary book, ignore me completely. I'm just making an ad-hoc suggestion how I'd do it.
I was thinking something along those lines too. but I have some questions.
Where does the 24 points of damage come from? did you just pick a suitably big number or do you have a method, that you did not share in your post?
It seems to me that the damage should scale to the situation. E.g. a strong giant competing with a strong spell would inflict more damage, than e.g. tearing a grog out of a tornado with a similar spell, as the tornado does not grip as tightly as a giant. Likewise yanking a staff from a person would likely cause less damage to the staff, no matter who wins, because a person has less power available with which to hold on very tightly than giant.
24 = 13 + 11 (the value of the opposed rolls in the hypothetical example).
I'd also use a contested roll, with two caveats:
A few spells already give strength values or easy factors, and I'd try to match this values whenever possible. It's not always clear how this is to be done. ReTe guideline 3 for example seems to pretty consistently have Str +5, but Hands of Grasping Earth uses the ReTe 3 guideline and has an EF of 15+ for the roll. The Treacherous Spear, on the other hand, calls for Strength stress rolls of 6+ every round, and uses ReHe guideline 10. In the end, some judgement call will be needed.
Based on how physics work I wouldn't sum the values of the opposed rolls, but use the lower of them to source the damage (think of the extreme case: if you are pulling something and the opposed strength is 0, than the thing shouldn't take any damage)
I'm also not aware of any rules for this in supplementary books.
A related question is: what happens if two magae target the same grog with two MuCo(An) spells, one turning him into a wolf and and the other into an eagle?
One spell will go first and turn him into an animal, the second spell will then turn them into the other type of animal. (Humans turned into animals by spells are still affected by Corpus effects, otherwise the second spell would have fizzled due to not having a proper target.)
The problem is that MuCo spells sustain the change - if they cease, the target reverts to his true form. So the second spell still has to "fight" the first spell.
A wogle, or an eagolf?
To be serious, I agree with Erik T. Resolve the spells sequentially. While yes the first spell is sustained, the second spell is over the top of it.
Let's think of another situation. Say there's an aura of rightful authority, and then he's turned in to a frog. The aura is still there being sustained but the second spell overwhelms it. What authority can a frog have, and even if we say the spell could work, how does one obey "Griddup"?
If two spells cant work at the same time, I would generally rule the second spell overwhelms the first.
I sort of agree. I would think if the successful opposed roll was 5 or more (say 3 if the SG is running a kinder world), one could make the damage half of the lower roll, as it's not a tug of war. It's either a fast yoink (the technical olde english term*), target ripped from the enemies clutches, or a case of the Giant holding the grog hardly needing to exert themselves.
For a legit tug of war, I think the summation would be better. I accept that's brutal, however they've got a grog grabbed by a giant, so I think this is a world where death is a real risk. At any point some foe can roll 1, then 1, then 10 and kill, so AM there's always that death risk no matter how careful one is, however, a giant with a decent roll can 1 shot kill a character without a bunch of doubling.
*I'm sure there's linguistic experts on this forum, so, before I'm called out, it is not an olde english word, it's a joke.
Somehow opposed strength rolls do not capture the feeling I'm looking for with Ars Magica. The giant would use strength of course, but it would be nice if the magus used the power of their magic in opposition.
A consistent mechanic for the force of spells seems to be missing in Ars. I can imagine that each form has different levels of force if the art can effect the physical world. Terram spells would have more brute force/level than auram ones. Added to this could be the casting roll. Rolling higher than the level of the spell would allow a magus to exert more force if they want to (perhaps having to choose between using their casting roll for either penetration or increased force)?
Food for thought anyway.
I'd be wary of that, because increasing the power or effects of the spell (including increases in strength for moving objects) would fall under the purview of Wizard's Boost. For another similar examples, if you want to raise the damage that a projectile will cause you don't raise the casting total, you raise the spell level.
I think you could somehow try to use finesse to make the target slip by the giant's grasp (even avoiding some or all of the damage), if the spell was originally designed to allow for fine control of the target's movement (and in that case I think the opposed roll of the giant would be quickness). Outside of that, the baseline (with eventual increases in magnitude) defines the total strength that an object can exert.
What would that be, exactly? From your initial question, it seems to be a straightforward case of opposed strength rolls between the spell and whatever is resisting it.
In the subject of related questions: assuming that the giant has MR, do you need to penetrate to affect the grog? I would go with yes (since you'd also need to penetrate to affect the giant's clothes, or a club he is holding).
If the only MR that counts is the grappler as the grappled counts as an object, it means a magi once grappled by someone or something with no MR has their MR removed. I think a living object must always maintain their own personal MR.
That's why I said a grog. If targeting a grappled magus, yes, I understand that you would have to penetrate both the giant's MR and the magus Parma.