Who were men that courted Elizabeth I after she was queen, specifically, who were her favorites and how intimate, if at all, were their relationships?
I've listed Elizabeth's various suitors at http://www.elizabethfiles.com/info/elizabeth-is-suitors/, there were quite a few!
The ones who got close to Elizabeth are:-
Thomas Seymour - When the teenage Elizabeth was living with him and Catherine Parr, he flirted with her and "abused" her.
Robert Dudley - I think he and Elizabeth truly loved each other and would have married if she had not been queen. I do not believe that they had a sexual relationship though. http://www.elizabethfiles.com/info/bios/robert-dudley/
Robert Devereux - He was a favourite later in Elizabeth's life. http://www.elizabethfiles.com/robert-devereux-2nd-earl-of-essex/4474/
François, Duke of Anjou and Alençon - Elizabeth affectionately called him her "frog" - http://www.elizabethfiles.com/on-monsieurs-departure/3873/
Sir Christopher Hatton - Elizabeth called him her "mouton" http://www.elizabethfiles.com/christopher-hatton/4533/