Your mum honestly can't sleep in with you. It is illegal to squat. Besides, the single-sized bed is too small for the both of you. Are you going back on the weekends? If yes, it will probably ease her anxiety a bit better.
All halls are good halls. Seriously. It depends on what you want as each hall has some minimum facilities. The oldest halls don't have the best facilities but have the best price, Hall 7 is very ulu but in walking distance to SPMS and The Hive (SSS/SoH), so most of the students I know are from these schools. There are the sports halls, like Hall 16, where they may request you move over if you are in the school team to play for their hall. Then there are the more kampong-halls, Hall 8-11 where everything is more chill. And there's the newer halls, best in facilities, but not the most established in CCAs. Anyway, as a freshie you can't pick your hall, and a good number I know got very attached to their hall and didn't want to change halls after that.
Go for your hall FOC! You can skip all your major/union FOCs but not your hall FOC if you want a somewhat fulfiling hall life.
Yes, NTU has made it clear this year it should be open to all. The choosing part, I can't divulge much but don't worry if yours is private. GLs pick so that they get some more hyper campers to make up for the less high campers. If you go with an open mind and participate actively, it should be fun.