Many hosts will be able to accomodate your needs.
You could either choose a free forum or a commercial version such as vbulletin (this forum is vbulletin)
Similar ones though that are free are:
www.phpbb.com
www.invisionboard.com
Your host would need to support PHP, MySQL (and most hosts will support this as its pretty standard now)
Either of these boards can be installed on your account in less than 10 minutes.
As far as a domain name, sure you can register any name you wish (as long as its available) If your site will not have any other content then something like myforumname.com
would work well. I think you would be safe in just ordering a small account for starters at around 50MB and 5GB transfer.
We have people using smaller 30mb as well and have forums.
Just send some pre-sales emails to your potential host with questions as you may need some assistance along the way.
Hope this helps
