tweet.IM is working great within my GTalk client software on Windows.  However, the messages are not being received by the GTalk client embedded in gmail.  They are also not being received by my Blackberry GTalk client.

Could this have anything to do with the fact that the service appears offline all the time?

It may be the case - I’m using tweet.IM with native Gmail account and the tweet contact is always online and works fine.

What you can do is the following:
1. Sing out of tweet.IM using the sing-out form in the FAQ - https://www.tweet.im/content/signout/
2. Delete any remaining tweet.IM contacts from GTalk client software.
3. Go to GMail and delete any remaining tweet.IM contacts from your contacts, like in the attached image.
4. Disconnect from GTalk on any clients, and connect again.

Now, go to tweet.IM and signup for the service again.

This routing should help, tweet.IM works great on GTalk for me (as opposed to Jabber, when the contact goes often offline) but if you still have problems, just post on the forums.