Moving

Okay so I've moved 2 times since my daughter was born. She's about to be 5. 1st time is when I moved out of my parents house when she was 11 months old, 2nd was moving from one apartment to the one next door when she was almost 2 because it was bigger and I was pregnant with my son so we needed the extra room. Well I'm thinking about moving again and someone told my EX (daughters bio dad but hasn't seen her since she was 5 months old). Apparently he said he's taking me to court for full custody because I'm not stable and moving around to much. I have full custody and he's never seen her or paid child support. Can he do that? Also I said I *might* be moving how would yall feel moving next door to your mother in law? They have a trailer that they're allowing us to pay $500 a month (our rent currently for an apartment is $800 and goes up every year) for 2 years and then we own it. I've never lived in a trailer so I don't know how to feel about it. Its 3 bedrooms and his step dad just remodel the whole thing. Any pros or cons?