1. Not a fan. I don't think it sounds pretty.
2.
Love it! I'd rather see this on a real person. :)
3. Tavey or Tavia are my NNs of choice, though I'd call her by her full name most of the time. I prn Tavey the way I think you prn Tavvy, but I don't like the spelling you've used. It goes against the norms of linguistics, but that's me being a nerd.