There are many types of honeysuckle. Some are safe to eat, others are toxic or have toxic parts. Unless you are very good at plant identification, I'd err on the side of caution.
Kudzu is, apparently, quite tasty for goats and rabbits and such. BUT, if even the smallest piece makes it's way to your soil, it will eat your house. If you have it already, then by all means, let the goats and rabbits eat to their hearts content...that will at least slow the monster down!
I believe that Scarlet Runner Beans are an edible vine that rabbits (wild ones at least) eat with great gusto. Very pretty on a fence or trellis, to.
Edited to add: Scarlet Runner Bean is Phaseolus coccineus.