Hello Hela, how are you?
Your sentences and interpretations are fine, for BrE at least, except that you need to use the indicative:
1) I am to tidy up my room before my mother gets back.
= I will have to do it (obligation given my an external authority)
2) I was to tidy up my room before my mother got back.
= I had to do it (but the interlocutor doesn't know if the speaker did tidy up his room or not)
3) I was to have tidied up my room before my mother got back.
= I had to do it but I didn't.
It's possible that some speakers would use #3 with the sense of #2; but I think your interpretation would hold in most cases.
Student or Learner