Is Skyrim Nexus (mod SDK) custom made? - Yes. It was made specifically for TESV Skyrim And you are not allowed to take any models from there unless you are going to use it with the game itself, or give it into the official modding society. Otherwise it would be considered stealing intelectual property/breaking the EULA you accepted upon installation of the software.
Mods are MODIFICATIONS made to some kind of game. So 99% of times if there is a mod SDK it is created by the company, or by the game fans and they rely completely on the game content + are not allowed to distribute/resell anycontent they get their hands on.
Is it possible to use the models in your published game? - NO
Can a modding SDK be made? Yes, as long as you have a +-finished game, otherwise it wouldn't be a modding sdk. But just an SDK.
Porting 3d "models" between sdks: the models are compressed in the Data folder (almost all the time). And they are saved in some format for eg. .FBX or .OBJ once you can export those outside the model container you can convert them into a model that the MOD kit supports.