A normal-size balloon of helium can lift about 10 grams. Hares can be about 4 kilograms. Totally doable but that's a lot of time, balloons and helium to just waste!
Anyway, here are some more species and approximate adult weight when alive:
That's why I'm listing the live weight. A dried-up rabbit carcass partially eaten by a fox will of course weigh less by a highly variable fraction, whose estimate I'm leaving up to you.
Probably not, the cluster of balloons would be so big you'd need a considerable amount of string, and at some point the overhead will be so big that each balloon will add more weight than lift.
Still, a moose is probably possible, just with more balloons.
You can't infinitely chain balloons because the string has limited tensile strength. There indeed is a limit but it's pretty high, especially if you use a fishing line.