How to address email to a professor who prefers informality?

If the professor explicitly says not to call him by his last name, then don't. It would actually be more impolite to ignore his request in order to conform to some abstract "formality" rules.

However, even if you are addressing your professor by his first name, you can still preserve some degree of formality. Some common norms include:

• do not use slang
• avoid excessive contractions (e.g. use have not instead of haven't)
• use your full name in the signature
• be polite

In general, the rule of thumb is to be respectful and professional in your communications. Sometimes that includes addressing the person as "Professor X" or "Dr. X", but it does not have to. Just make sure you sound polite!

Rules exist to serve people, not people to serve rules. The default is formality, because if you don't know how someone wants to be called, then formality shows respect. If you do know how someone wants to be called, them calling them that shows respect. Don't confuse the means for the end. Formality is the means, respect is the end.

I would suggest that you use the name/term/concept that the professor signs his initial reply to you with as the introduction to follow up e-mails. This approach ensures that you are always correct and formal in your initial contact and can be informal in the subsequent contacts if the reply allowed you to do so. See the following e-mail headers/footers as an example:

Original Contact

Dear Prof. FancyPants,

...

Kind Regards,
Hopeful Grad Student

First Reply

Dear Hopeful Grad Student,

...

Cheers,
John

Your Reply

Hi John,

...

Cheers,
Andy

I hope that the above illustrates how I would treat your situation.