I have bought a domain name with the word Facebook in it. Can I use this domain name for a Facebook chat application that I own? I also use adsense as my main ads revenue, would this violate their terms? Any help would be very much appreciated. Thank you.
the short answer is yes, it is a violation of facebook's registered trademark name, they could ask you to not to use the domain and if ignored they could take you to court over it
Would it make a difference that the domain name mentioned will forward all traffic to a Facebook app? This would mean that all traffic created by the domain name will be sent straight to Facebook.
No it doesn't - it's still trademark infringement. You need express permission from the trademark holder, and they usually don't grant it.