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So far as domain names are concerned, you are free to register any available names that you want, however, if you attempt to misrepresent a website as being official, authorised or affiliated, etc. then you'll be in trouble.
Similarly, if you register in "bad faith" (e.g. with the intention of selling to the owner of a similar domain name or trademark) you'll be in trouble.
Finally, if you register a domain name that someone else uses as a trademark or if you register a domain name that is someone else's official company name, you could be in trouble. However, in these cases, it would depend on a multitude of factors and you would need a lawyer who is expert in such matters - such advice cannot be given here.
1) Claiming copyrights for a domain name probably wouldn't hold up in any court of law in the world, unless it was one REALLLLLYYYYY long domain name.
2) I tried to register a domain as a trademark once and got rejected because the domain name consisted of two English words stuck together with no space and a .com, which basically made it an address and that you can't trademark common words.
As long as you didn't register the domain name in bad faith and the root isn't a trademarked string of characters (e.g. Google) I wouldn't loose much sleep over it. Just don't try to sell it to the party or they will get you for a bad faith registration. Remember trademarks are not universal, they are for specific industries.
also I thought that you can copyright a name as long as its not a generic word(s).
You can CLAIM a copyright for anything you want, but whether that copyright is valid or not is another story. You would be hard pressed to convince a court that a domain name deserved copyright protection because it it is too short and not really a unique piece of creative works. See: [chillingeffects.org...]
Question: Does copyright protect words or short phrases?
Answer: No. Names, titles, and short phrases are not subject to copyright protection. These are not deemed to be "original works of authorship" under the Copyright Act. Names may be protected by trademark, in some instances. See the Trademark FAQ for more information.