What we have is "imagined past", which is a past modal + have + past participle. It tells us that the action definitely did not happen in the past.
"I should've trusted the boss man" means you didn't trust him, and now you regret it.
Yes, auxiliary verbs can take contractions. You already know this with negatives (hasn't, isn't, didn't, won't...) and we can do the same with the past modals: should've, could've, would've, might've.
Yes, we definitely use it!!