is it a crime to log into someone else's email? Is there a federal cyber crime statute which make unauthorized access to an email account a crime?

Hacking an email account is a federal crime under The Electronic Communications Privacy Act (ECPA) which sets out the provisions for access, use, disclosure, interception and privacy protections of electronic communications. ECPA was enacted in 1986 and covers various forms of wire and electronic communications. According to the U.S. Code, electronic communications "means any transfer of signs, signals, writing, images, sounds, data, or intelligence of any nature transmitted in whole or in part by a wire, radio, electromagnetic, photo electronic or photo optical system that affects interstate or foreign commerce." ECPA prohibits unlawful access and certain disclosures of communication contents. Additionally, the law prevents government entities from requiring disclosure of electronic communications from a provider without proper procedure. The Legal Institute provides Title 18 of the U.S. Code, which encompasses ECPA.

Sure, it's a crime to use someone Else's email address. While the legality of accessing another person's email may be questionable, using someone else's email address falls under the Identity Theft and Assumption Deterrence Act.

Its a crime to use personal identity of a person by using his personal account,it can be email,fb,twitter aor any account you are violating ECPA.

How can they prove that you accessed someone else's email? Is there some footprint of the computer's ID number on the email account?

They can tell if you've sent someone an email based on your ISP records, the mail headers of the received email in the recipient's inbox and also by recovering any deleted (and obviously non-deleted) mail files you have on your own computer if you use a client like Outlook.
They can tell if you've accessed someone's email by IP logs of that account that the provider will keep and again, by your own ISP records. If you connected through a proxy or some other way of hiding your online identity, they would have to find a way of showing you connected through that. Really, if you did it from your home, they can prove it no matter what you connected through.