Everybody develops their own preference on hosts, no surprise there, there are hundreds if not thousands of capable hosts out there right now. Between my day job and my freelance work I currently have sites on 9 or 10 different shared hosts, just depends on what technologies your site needs to run efficiently.
Godaddy does do hosting. Since it sounds like you did straight up HTML, their "economy" hosting will probably handle what you need at a reasonable price ($4/mo). It may be nice for you to keep the registrar and host as the same. If you want something even cheaper they're out there.
As far as the trademark and copyright stuff goes always mark what you legally can. It won't stop someone on the internet from using it if they really want to, but it will provide some recourse if you find out about it using it, especially if it is registered ®copyrights © and unregistered trademarks are much harder to protect.