When the technican comes out he can run you a new cat5 home run to your new uverse modem, so if you are nice and run one before he/she arrives this would be the min telephone wire for uverse. The existing coax can be used for hooking up the stb(s) you will need for each tv. The tech will test them to make sure they are not losing more signal than they should and should change every fitting to a blue att compression fitting including any fittings behind wallplates and the wallplates themselves or the barrel in the wallplate.
The 250gb cab shouldnt include tv as far as I know.
Im not sure about double nat or dmz since I havent set those up or had any customers ask about them before. Someone should be able to address that. You can open ports on the router for xbox live and gaming that I believe may be some of the problems people have with xbox live and uverse
Uverse only allows a self install for internet only, so if you switch to the uverse for internet only you could do it yourself but tv always requires an install visit plus you get education on how it works
Our tv service requires all of the coax to be tested and all the coax has to have our fittings on it as well as our splitter so thats another reason for full tech installs for video