Its not straight forward to answer Mad, but I will try.
First things first, 1and1 - I would recommend them, Will on the other hand would not.
If you do decide to go with them, I have an affiliate link which earns me a bit of commission for recommending them. That said, I wouldn't recommend them if they were rubbish.
2nd, many hosts will not allow chat channels via mysql as the load is too great. They will either say no straight off (as 1and1 do) or they will let you run it, before pulling the plug without warning when it puts too great a load on their myqsql server.
3rd, forum licenses. To run 6 forums with different VISIBLE content requires 6 licesnses (even if they are on the same db, but with masked forums for each URL). You may also get cookie issues with the login as each URL will require a different cookie.
4th, in the scenario you quote, they would use the same database for the login, but the chat would have its own tables within the db.
5th not sure about the legality or etheics of joining a person to 6 seperate forums in 1 go - they may not want to and may see it as a breach of their privacy.
In my experience, it wouldn't work in terms of its cost or practicality and once one forum gets popular, you are going to be into bandwidth issues.
....other than that, its a good idea - lol