Does the skin effect and the proximity effect cancel each other out? I'm interested in how the Skin Effect and the Proximity Effect interact with each other. 
From what I can understand:


*

*The Skin Effect is when AC current 'collects' on the skin of
conductors due to the counter-emf from its magnetic field. 

*The Proximity Effect is when the magnetic field of a  wire induces eddy currents in adjacent wires, which 'push' the current away from conductors carrying current in the same direction.


Now in the case of a transformer, the wires are close enough for the Proximity Effect to occur, so could the two effects average each other out (to create a uniform/close-to-uniform current density)?
Thanks in advance.
 A: No. The skin effect pushes the current towards the surface of the conductor.
The proximity effect acts in your case in such a way to push the current of the other turn of coil away from the other wire that is inducing emf in it. That means that the current in the wires will still be concentrated around the skin of the conductor, but it will be more at the side away from the conductor and a bit less in the side near the other conductor.
Now this is only for 2 turns of a coil. If you have a long coil, then the effects will kinnda average out because the wires next to a middle turn have the exact opposite effect on it.
EDIT: To add to this answer, If you have multiple turns of coil, then the current Will be pushed mostly to the inner side of the inner turn. This is also the location of most noise because the magnetic field there is also the highest and the current density as well. 
A: No, proximity effect and skin effect are two different things. Due to skin effect, AC current tend to flow on the outer surface od conductor. This is due to linkage of its won magnetic flux due to which the inductance of inner strands becomes high and hence do not allow current to flow from the inner strands.
While proximity effect is the result of magnetic flux linkage of nearby conductor. Because of this, the current distribution on the surfcae becomes non-uniform.
