e-mailing a professor: first lines of an email (US/UK)
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.