Thrust happens when there is some expulsion of a fluid (such as gas) in the opposite direction of motion. Thrust is used to propel aircrafts forward and thrust can be used in rockets to overcome the force of gravity. Planes achieve it through the combustion of fuel.
I'm not sure what exactly you mean here. If you are referring to Newton's third law, then the force equal in magnitude, but opposite in direction to the force exerted by the gas onto the plane would be the force that the plane exerts on the gas.
Lift refers to the force that opposes the weight of the airplane when the plane is in motion, so weight is earth's gravity affecting the plane.
An airfoil provides lift by the way it is designed. When an airplane takes off, the air can flow above and below the wing. The air flowing above the wing is traveling faster than the air travelling below it. This cause the pressure above the wing to be lower than the pressure below the wing. Due to this difference in pressure, a force will be provided in the direction from higher pressure to lower. This creates "lift" and thus, the airplane may take off once lift has overcome gravity