What are the relations between the level of power and abilities and the promotion of Pawn?
No, the ability doesn't depend on the Queen of the group. I forgot whether it was Rias herself, or Azazel-sensei that said that the stronger Issei is, the longer he can stay in promotion as the promotion takes its toll on his body. If the promotion depends on the power of the team's Queen, then there won't be any need for Issei to improve his physical strength.

In the light novel, it was explained that each piece has a value. Queen worth 9 pawn, Rook worth 5, Bishop and Knight 2. So, rather than the strength of the Queen, the promotion increases the strength according to those numbers, I think. Of course the improvement depends on which piece he's going to promote into.

For example, Knight is worth 2 Pawns, so the ability improvement is more or less 2x of his ability as Pawn. However, it is not distributed evenly. Most of it will be distributed to the Speed. Queen, being all-round piece, will get an even increment to all abilities.