[sp332]

Set up email forwarding. Sometimes even if they suspend your account, new emails will still be forwarded. Also, make a backup of your current emails by setting up a desktop client (e.g. Thunderbird) over IMAP and let it download all your current emails.

As for proof: saved web pages, screenshots, copies of any communication you have would be helpful. There was one case I know of when someone at Google was spying on users' email, they took the case very seriously and fired him.

As for whether they have a right to: I'm sure they do. Anyone running any email server has the right to look at your emails for purposes of maintenance etc. From their privacy policy: We restrict access to personal information to Google employees, contractors and agents who need to know that information in order to process it for us, and who are subject to strict contractual confidentiality obligations and may be disciplined or terminated if they fail to meet these obligations.