I'll say the below, even though moderation issues are not generally discussed (as per the rules, if you've checked the Policies thread...). However, anyone these days that is familiar at all with the way the internet works (back end) will already know this, so I'll state the following as a one time thing:
Yes, it happens often. Yes/No, proxies and dynamic IPs are very common. I (personally) don't seek out anyone, unless they're trolling/being a jerk/dummy and/or causing problems.
Bans are enforced via IP, generally.
Bans are not always permanent, but we (I and the TW Staff) don't publish publicly any of this info. The user that is banned is generally told how long.
Temp bans are generally a week or two.
If a user comes back within that time period, they are permanently banned (take the time out "like a man," learn a lesson from it, and don't make the same mistake twice).
Here's my (personal) point of view from a moderator standpoint (I'm not speaking for the TW Staff).
If you understand why you were banned, have made changes (and/or are willing to change), I'm cool with it. I personally would be happy lifting a permanent ban IF a user has made a real change.
With that said: If you're willing to acknowledge the errors of your ways, and move forward, the BEST method of return is to actually contact the TW Staff, state this, and we can all move forward. It is worse for everyone if you decide to return under an alternate ID, because people legitimately browsing under a proxy are put at risk for undeserved bans. If you're willing to "man up" (I'm using this term as a figure of speech, and don't intend to offend anyone), contact the admins, most of them are flexible, and willing to give a second chance.
In general, though, the people that get banned are PURPOSEFULLY trolling, and are having "fun" at the expense of others. Quite frankly, if someone was trolling hardcore, and wanted to come back and be a "regular ol' " poster, I would have 0% problem with it. They are welcome if they want to contribute to the community as a whole. This is rarely the case, though, as internet trolls actually get joy from making others upset.
As far as whether we're aware of repeat/return offenders, I'll say this:
IF you decide to surf this website through a proxy service, you are EXTREMELY likely to be sharing at least one previous IP address with another user. This is a bannable offense. Don't give us a reason to look into it. Use a proxy at your own risk. Of course, there are legitimate reasons to surf the web through a proxy, so we're not casting a wider net than necessary.
For instance... there is AT LEAST one user in this thread that has a shared IP with another user
If I'm given a reason to investigate said user (AKA they are trolling, or are frequently getting reported for rule violations, etc) they'll be gone the same day. Capiche?
-Your friendly neighborhood moderator