Learn Italian: Lei disprezza quello che avevamo.

Chi is strictly used for people, meaning whoever or the person who. Because this sentence refers to a thing, meaning the thing that we had, you must use quello che.

Can I just use cosa to mean what in this statement?

No. In statements where what means the thing that, you must use a compound relative pronoun like quello che or ciò che. You only use cosa or che cosa for what when you are asking a direct or indirect question.

Could I use ciò che instead of quello che?

Yes! Lei disprezza ciò che avevamo is perfectly correct. Both phrases mean that which or the things that. Quello che is more common in everyday conversation, while ciò che sounds slightly more formal.

Why is avevamo in the imperfect tense rather than the passato prossimo?

The imperfect tense is used here because it describes an ongoing state in the past. What they had (their relationship or situation) existed over an unspecified period of time, rather than being a single, completed event.