In which cases (or how) can a user become anonymous on an SE site?
If you know you want to be anonymous before you post, you simply log out if you're logged in, and ask a question as an unregistered user. An email address is required, but you can use any number of anonymous email services to provide you with whatever anonymity you require. Proxies, Tor, and other services can provide further anonymity.
Some users have pseudonym accounts. While "sockpuppet" accounts are discouraged, multiple accounts which do not interact or otherwise abuse the system are generally acceptable when privacy concerns occur.
If you've posted material that you need to disassociate from, you can only email the stack exchange team and request that the items be disassociated with your account, or your account deleted. Typically the items themselves will not be deleted, but if you have a particularly unusual, compelling case, they may be willing to do that for you as well.
Are there any levels of anonymity on SE sites?
Not really. Unregistered users, I suppose, could be considered more anonymous than registered users. But there's no formal system or hierarchy of anonymity.
Can a registered user be anonymous as well?
Only by posting while not signed in, or under a pseudonym.
Can I have a normal profile on one SE site & an anonymous profile on another SE site using a single SE account (or email)?
Your accounts would be linked by Stack Exchange, and would be visible to people on either site by going to your network profile. You will need to use different OpenIDs for each account to avoid account linking.
While you may be able to create two separate accounts with different OpenIDs, but the same email, they may be merged at some point in the future. for a variety of reasons. If you need anonymity and a registered account for each, your best bet is to use two different accounts with different email addresses.