With the exception of free-file sharing sites, I would think that any subscription-based service wouldn't allow you to login from more than one computer. Otherwise any number of people could use the same account from anywhere, which obviously would have a negative impact on the provider. It can be a hassle though if you have the jobless roomie competing with you for listening time!
As for burning the tracks, I'm still not quite sure why it wouldn't work for you on Rhapsody the way it would work anywhere else. If you have all the tracks you want in your playlist, you can select any number of them in any order and burn them to CD. The only problem you might have is if there happens to be a track or two you wanted in the same compilation that isn't available on Rhapsody. But then, like you said, you just put all the ones you can get onto CD and then get the other tracks somewhere else and rearrange your compilations later if absolutely necessary.
In a perfect world, they would have every single track ever recorded, but 350,000+ isn't a bad start if you ask me. Sometimes they don't have something that I really want to listen to, but it's the exception rather than the rule. I still highly recommend it to anyone who hasn't tried one of these before. In fact, I'm sure there are people here who have no idea what we're talking about, so for their edification... -->
http://www.bestbuy.com/rhapsody