Every institution is going to be different, however, I would say that in general few academic institutions pay extra to be a mentor. This may be different at community programs, although at the few community programs I am aware of their workings, only the program director and a few others get anything extra, and some of the private attendings actually pay into the program in return for the night coverage that their patients get).
As far as what your mentor does for you. This definitely varies. Some mentors are very hands on, some are just there if you call them, some are totally absent (this occurs mostly, when you have one assigned to you I would guess/hope). The key is to choose a mentor wisely. I would say that your 1st priority is to look for someone who is going to look out for YOUR interests. This should involve him/her understanding what your goals are in life and helping you figure out how to achieve them. In general, they should hopefully share your outlook on life and have a compatible personality to yours, or one that you can deal with. Often, this will mean that you find someone in your expected field and often you will find yourself doing research with them. This is certainly not mandatory. If you have chosen wisely, they will recognize if this is the right move for you or not. If you end up with someone outside of your intended field of specialty, you should look to find a secondary mentorship with someone in the field that can give you give advice about that field specifically and assist in your fellowship search. No one says you can't have 2 or 3 or more mentors.
I personally have multiple mentors: 1 for my research interests, 1 for my specialty interest, and multiple others that I go to when I need other types of advice that I respect their opinions.
Last bit of advice, make sure to find someone who will give you the straight truth when needed. Cheerleading has it's place, but you also need someone to tell you when you are screwing up, going down the wrong path, or not competitive for certain fellowships. This can keep you from making a huge mistake every once in a while.
