I don't and personally, I don't think I have ever heard it used in that way. However, I have heard "honey", "honey bun", "duck", "pet, and "love". However, other people's experiences may be different.

There is also the word “bonnie/bonny” (ˈbɒni) which could be mistaken for “bunny” (ˈbʌni) when spoken or sung. There is the old song “My Bonnie Lies over the Ocean” in which “bonnie” refers to a loved one.

No, and I've never been called "bunny," either. However, terms of endearment can be regional, or (in the United States) associated with an ethnic group, or a family tradition... or just one person's own style or manner. There's a great book, a classic for reading to little children, entitled "The Runaway Bunny," by Margaret Wise Brown. A little bunny tells her mother various ways he is going to run away--"I will become a fish in a trout stream and swim away from you," and the mother replies "If you become a fish in a trout stream, I will become a fisherman and I will catch you." I can easily imagine that in a family where the kids liked that book, "you are my little bunny" could become a term of endearment within the family.