"are there any free DNS registers to host my own DNS server?"
That statement confuses me. If you are talking about dns server software then yes there are a few about.
Both www.zonedit.com and www.granitecanyon.com are free to use(zonedit is restricted in a few ways only 5 sites no more than 1 million dns requests). I have used zonedit.com extensively and they opperate a great service, although update times can be slightly slow some times; caching is too long.
The catch with running your own dns server is that you need 2 or more of them, most if not all domain registration companies insist on it. There are schemes to swap secondary dns servers though.
If i were you i would make sure you get a static ip address for your internet line (makes life a lot easier) and then use zonedit.com for all your dns needs.
If you want to find out more information on setting up a webserver from home visit www.dslwebserver.com.