Legal Question in Intellectual Property in California

Probably; it's done widely. Things to avoid are passing yourself off as the real thing, using copyrighted lyrics, etc.

It depends on the celebrity, and whether you're using that as a means of promoting the song. If you're just doing a few bars as the celebrity and it's clearly meant to be an impersonation, it's probably okay. If you mean using the celebrity's name and image as part of your publicity campaign, you'd better check with a lawyer knowledgeable about right of publicity (and not every lawyer claiming to practice in entertainment is). The laws vary in each state and it depends on the celebrity's legal residence at the time of death.