If you consider the grid fins at a small angle of attack, am I right that drag will act in the direction of the flattened axis (thickness of the grid of fins) and that lift will act perpendicular to it?
For any body in a flow, you can draw a single net aerodynamic force vector. Conventionally, the component of this vector inline with the vehicles longitudinal axis is called drag, and the component inline with the vehicles vertical axis is called lift. For the Falcon 9, the horizontal axis would be the line passing through the engines and the tip; this convention allows drag to be in the opposite direction of velocity.
Do lift and drag both contribute to the steering force on the rocket? Does one dominate over the other or are they both significant?
Yes, both lift and drag contribute to the steering. Consider if only one grid fin was actuated. Net force on the fin is increased which increases both lift and drag. The lift will create a roll moment (and a yaw/pitch moment), and the drag will create a pure yaw/pitch moment.
I would wager that the F9 control scheme has opposite fins deflecting at the same angle more often than not to cancel the moment created by drag. After all, you can get roll, yaw, and pitch from just combinations of lift from the fins.
Worth mentioning that, for considering the surface area of the fin creating the aerodynamic forces, only the grid strips inline with the rotating axel should be considered. The others will remain at zero AoA when the fin is rotated.