If בית המשפט means "court", how do you say "courtroom"?
אולם בית המשפט
Literally "the court hall"
what is the difference between בית משפט and בית דין ?
In the state of Israel בית דין is used for special court for specific cases, as traffic, family issues, etc. while בית משפט is the general court where more "serious" cases are being held.
Why can't the solution also be "The wife goes to the court"?
On its own, אישה doesn't mean wife, it means woman. You can either specify whose wife and then אישה fits (for example, "his wife" = אישתו) or you can use another word, like רעיה or בת זוג.
The lady goes to the courtroom was wrong...how do you say 'lady' then?