Hey there, figured I would try to help with these questions.
1. Its is indeed possibly to both serve the Citadel and the Wardens, Celeste Arumen is both an Apprentice and an initiate however it is very rare. The Citadel is not fond of their investments risking their lives in the Testing Grounds. As far as I know, Wardens work fairly closely with their initiates to train them up and make sure they don't end up dying horribly. I would assume that, like many denizens of Sahova most of their time is taken up by their work.
2. No, unless the Warden or Initiate are also a Wizard of rank in the Citadel they do not receive grants. The Wardens are completely separate from the Sahovan Wizards as an organization.
3. It probably depends on the Wardens, for instance the Warden of the Heartlands rarely leaves her domain. Some Wardens might communicate with others more, some might completely isolate themselves.
4. A number of Initiates are taken from the failures of Judgments, so I would assume that as long as you have a magic specific to that Warden they might take you. Novice or not.
5. I think Mirage is probably working on Eli, the Bloodhills Warden but currently there is no write up on him. I do know the lore states he specializes in a combination of Projection and Voiding.
Hope that helped, and if this info is wrong I'm certain Mirage can clear it up.