e-mailing a professor: first lines of an email (US/UK) Not being a native speaker of English, I would like to know how to best introduce myself, what sounds too formal, and what is maybe a bit too colloquial. I have a few questions I want to ask, but I am not sure how to start the email in the first place.

Since many people have pointed out the culture dependency of the question, I would like to make clear that I am concerned with the US / UK, or roughly the English speaking academic world.

Also, for future, subsequent emails, can I just start the email with things like:

Dear John,
How are you? ... 

Dear John,
I hope all is well in Brighton. ...

Dear John,
I hope you are doing good! ...
For the first communication, address the person as "Prof. [Lastname]" and keep doing so until the person says something like "Please call me John."
In the US, a common first liner for initial communication is "I hope this e-mail finds you well."
"I hope you are doing good!" should be "I hope you are doing well." Try to suppress the use of exclamation marks and save them for things that are really worth exclaiming about.
As for subsequent e-mails, what you're saying in the main message is a lot more important. A "Dear Prof. [Last name]" followed by a succinct statement about the main points would usually be sufficient as the start.