Depends. It basically depends on the extent of the work you do for her. You're basically doing labour for nothing. Then again, $3000 is a lot.

Uhm. Yeah. She just sounds jealous. Thats it. So it's normal.

You either cut to the chase and ask her directly "Do you want to go out and when" or move on. Don't waste your time with people who will eventually waste yours.

I'm very self-conscious about my appearance. I almost shower everyday now. But I use to shower once every week until I realised I smelt bad.