Can a bad breaker still allow the outlet to test hot?

Depends on what you mean by bad.
Bad as in will not trip, then yes.
Bad as in does not turn on, then no.

My dad worked on my panel and added a breaker to my panel. He tested the original line if 220 and it tested hot so we thought it was good until we installed the new air conditioner and it won't work... the breaker doesn't click when u try to reset it so um confused. (And yes I'm a girl and have no clue)

I'm confused too. What is the original line? And where did he test for power?

The simple answer is "YES". If you have other questions regarding this scenario, please elaborate on your question. Speculation by others will surely confuse you if you don't.