I have been trying to get a satisfying answer for myself for this question, but it has proven to be very hard.
For your first question there is no real name for them as far as I can see. I have looked at the episode (episode 2 around 4:00 minute mark) to see how the different subbers named them.
So as you can see one of them names it "animal furballs" and the other names it "pet balls". What info I can give you on it else is:
- The people from Biscotti turn into "Dog looking balls"
- The people from Gallete turn into "Cat looking balls"
- The people from Pastillage (season 2) turn into "Squirrel looking balls"
For the normal inhabitants can turn into the animal balls the following ways:
- When they have been hit hard with a weapon (KO/unconscious)
- When they get touched on the back or head
For your second question I have tried to look up information, but not much is said about it in the anime or is found on the web about it. The only thing I have been able to find about it is this:
It may be part of the protection magic. High-ranking fighters lose
their clothes. Everyone else turns into a furball. A pretty blatant
example of this happens in episode 9 of the first season. Vert Far
Breton shoots a magical arrow which hits Ricotta while she's flying
above them. Ricotta loses all of her clothes except her panties, and
then a magical spear hurled at Vert's troops knocks them all out, but
only strips her armor/clothing off.
Source: here under "Clothing Damage".
The third question is by far the easiest to answer. If you watch the second episode about at the same time mark I mentioned in the first question you will see them saying something in the lines of:
They will be rendered powerless for a limited time.
During this time that they are in their "ball form" the medics of the teams will "collect" them and bring them to the base camp of their team.