Is their any antidepressant that won't cause weight gain or affect heart rhythm/blood pressure?

Zoloft, Lexapro (escitalopram): Viibryd, (vilazodone hydrochloride) prozac, and Celexa at recommended doses should all be safe in this regard. Best wishes.

Yes, possibly: But there are also individual responses to consider. Wellbutrin (bupropion) is a possibility that comes to mind, but it will not help anxiety. (you list anxiety on your other information). Lexapro is relatively weight-neutral also, and does not affect bp, etc.