1. I'm not sure what the limits are , if any to using trainers with nGCD. But they work the same way seemingly to me as the regular game. I don't remember how nGCD handles trained skills, if differently. But the way you go about it is the same. I think tejon reccomends setting Progress's settings like this to work best with nGCD. And if you hadn't heard of progress, its really great to go along with nGCD, they used to be included together as nGCD itself.
set Progress.iLevelUpSkillCount to 9999
set Progress.iTrainingSkills to 9999
It prevents some trouble with using nGCD and progress that I don't remember but do remember that it is desirable to do.
2. There is a special ESP included with nGCD that makes the oghma infinitum work correctly. Details maybe available in one of tejon's threads or on tesnexus? I don't know, the current RC6 readme doesn't include tejon's usual very thorough readme files, but they are out there somewhere.
3.Its safest to install nGCD, like any mod, with either OBMM or the BASH installer tab in Wrye bash. It lets you know if the mod being installed will overwrite files that other mods use and possibly conflict with them in that way. nGCD replaces data\menus\levelup_menu.xml . I have had to make sure to install nGCD after other UI mods like darnified UI so that this menu will be replaced. OBMM and BASH (i guess?) will tell you that it is replacing this if another mod is also replacing the regular game's levelup_menu.xml . But that's my best guess, nGCD has been changing and its UI work has been improving a lot. It can show you your progress to levelup now too just like in the normal game.
3a. I don't know about the Grey prince's training, haven't tried it. But many things work well with nGCD. If it doesn't, tejon might like to know.
You may find more help by reading tejon's threads, search for his username by clicking members in the top right, and you can find those. Also there are other nGCD experts that hang around that may give you more information. Enjoy nGCD and remember it is highly customizable in the INI files.
Aren't I the adoring fan of nGCD. I don't have hair like that.