Yes, moderators can see your real name and email, although this information isn't something we see on your profile by default, we have to click several buttons to see it. We also have to agree to abide by the Stack Exchange Moderator AgreementStack Exchange Moderator Agreement when we become moderators. This states:
ii. I acknowledge that I may have access to potentially personally-identifying information about Ask Ubuntu users and that in connection with such access
a. I will use such information solely in accordance with the then-current Privacy Policy of Ask Ubuntu,
b. I will not disclose this information to anyone,
c. I will not store or copy this information and
d. I will only use such information in connection with performance as a Ask Ubuntu moderator for the benefit of Ask Ubuntu.
So you can be sure that your information is pretty safe.
As to your question about other things we can see, moderators have access to all privilegesall privileges as well as various other tools used in the moderation of the site. If you have any more specific questions ask away.