Should a student use the email account provided by the university to communicate with professors? I am an undergraduate and I do not know much about the rules of writing an email.
The email system used by our university is not convenient at all: it responds slowly, often crashes, and its common to miss important emails.

I need to email profs in our school and other universities.  Is it necessary for me to use the email account of the school? Can I use my personal gmail account? Is it informal or impolite?
In the cases I'm familiar with (U.S. universities), using your own e-mail account should be completely fine, subject to some obvious caveats.  One is that it's best to have an e-mail address that doesn't look foolish or offensive.  People sometimes choose very strange usernames, and you don't want that to reflect poorly on you; furthermore, you should make sure your e-mails include the name your professor knows you under.  Another issue is that if you are asking for sensitive information such as grades, your professor will likely be unwilling to send this information to an outside account without some verification that it belongs to you.  Finally, you should make things simple for anyone you correspond with.  For example, if you use several accounts, you should check them frequently or forward one to the other, so that there are no delays if someone sends something to an account you didn't expect.

One common solution is to set up your university account to forward to your private account, and to set up that account so you can send e-mail listing your university account as the sender/return address when necessary.