How do the 'Steeves' work? How do I tell which 'Steeve' is mine?

How do I tell which Steeve is mine?

Its AI will make him in general follow you while out of combat. So run other direction - Steeve should shortly follow, given enough distance.

In combat they will seek enemies, often going a big distance to another dwarf to protect him.

Tip: always try to tame stronger bugs (slasher and guard, see wiki) for coolness! They look cooler and are somewhat stronger (subjectively, never bothered to test it).

Is there a way to tell if my Steeve is still alive, especially if there are multiple Steeves running around after a fight (so that I'll know when I need to tame another Grunt)?

Apparently it doesn't matter. Steeve doesn't last long.

Just use your skill by cooldown on next bug to make minus one enemy. Kill another with charge pickaxe skill. Then back to destroying bugs in a classic way - with weapons.

Apparently, I can 'pet' an already tamed Grunt. This can get confusing if there are multiple identical Steeves in the vicinity. What does petting an already tamed Grunt do? Will petting another Dwarf's Steeve make that Steeve mine?

Petting does nothing, it's only for laugh. Petting may prevent you from using items (e.g. Molly), which is somewhat annoying at times.

Petting other dwarves Steeves is considered a crime. Be ready to take all consequences: angry dwarf and his angry friends may decide to shoot your down. If you are "petter" (that pervert type of dwarf who must pet pets of others), then do it when no one seeing you (during Dreadnaught fights is the best).

What happens to the old Steeve if I tame a new one? Does it become hostile again?

Then the old Steeve is gone. His soul will go to a better place and he will not be forgotten.. at least until you charm roughly ~100500 other Steeves, because it's proven to be impossible to remember that many at once.