Should I still use "Dear Prof." to begin an email to a professor that I know?
In the US, it is very unusual to combine the title "Professor" with a first name. I have never been referred to as "Professor Tom" in my whole career.

You can either write "Dear Prof. Smith", using the family name, or "Dear John". You should only write "Dear John" if you would call him "John" when you speak in person. (For people you haven't met in person, the decision is a bit different; but for people who you know personally, this is a good rule.) For example, my students generally call me "Prof. Church", but sometimes after they graduate I invite them to call me Tom; in this case, it would be perfectly appropriate to send me a letter beginning "Dear Tom".

I would use "Hi John" only to convey a very informal tone. Even with colleagues or students with whom I'm good friends outside of work, I still use "Dear Mary" when contacting them about professional business. For this reason, I don't recommend using "Hi Prof. Smith"; it mixes an informal term with a formal term.