Can the father of my baby call immigration on me if I file for child support?

You are potentially looking at a domestic violence situation. Speak to an immigration attorney right away. Unfortunately he can call them but whether he can get them to do anything, that's a whole different issue. If money/fees is an issue, seek help from your local legal aid society.

Unlikely, unless you are an enforcement priority (have committed crimes, fraud, previously deported etc). ICE generally isn't looking for single mothers who aren't enforcement priorities. I would also say that you have every right for child support. Consult with an immigration attorney to evaluate if you have relief and perhaps that attorney can help you with suing for child support as well or finding a family lawyer.