Does the library have rooms with computers so that anyone can use to do something in private away from the outside of closed doors?

They have laptops you can check out and use the private room

There are laptops and study rooms, both of which check out to your library card. There are two total study rooms and they are first come, first serve - and have a two hour time limit.

Yes they do. Just remember there are restrictions on certain sites which you won't be allowed to visit.