borysgo2,
Bearing in mind I only know what I've read about this, as I'm unable to really measure what's going on inside a wire in a transformer...
How thick a wire you use depends on how much difference between DC and AC resistance you design for. Theoretically, as long as the radius of the wire is less than the skin depth (and therefore the diameter is less than twice the skin depth), Rdc=Rac. The cross sectional area used at a given frequency might be less than at DC if the wire's more than twice the skin depth in diameter, and this decrerased area equates to a higher resistance and therefore a higher copper loss at that frequency, but depending on how much copper loss you can tolerate you might calculate that it's still OK. In an inductor in continous current mode such as the buck inductor for example, the DC current is greater than the AC current, so the AC resistance can be proportionally larger than the DC resistance and thicker wire used for a given frequency.
Proximity effect is where things get a bit different. In order to induce a current in a wire, a magnetic field has to be able to reach it. If it can only penetrate half way though a wire, it will induce a current only in that half of the wire. For a single layer of wire, that's not much of a problem. For multiple layers however, you might get the effect that the current running on the near side of a wire is partially opposed on the far side of the wire and you get little magnetic circuits set up between the layers. The net current through both will be as expected as the whole system is in equilibrium, but the copper loss will be greater than expected (e.g. net current through a wire is 2A, but 3A run through on one side in one direction, 1A in the side in the other other direction. The net current is still 2A, but the resistive loss is as if 4A runs through it). The more layers used, the exponentially worse it gets and you might see that this could potentially equate to a greater loss than simply the skin effect alone. That's why I use wire of the same diameter as the skin depth, but it's still a pretty arbitrary thickness mainly designed for my own peace of mind.
Here's an image courtesy of wikipedia of one method of calculating Rac/Rdc ratio for various thicknesses of square wire (h) compared to the skin depth (delta). You can see that the skin effect alone is shown in the light blue "1 layer" line. The proximity effect is shown in all the other lines. It looks to me like where the gradient of the line is twice as steep, that's where the proximity effect is dominant and where it's half as steep is where the skin effect is dominant. If you simply look up a vertical line down the centre of the plot at h/d=1 and look up how many layers of wire you have, you'll probably see that Rac/Rdc is quite high (by eye, at 5 layers, Rac/Rdc=4, at 10 layers, Rac/Rdc=16). Now if you look up h/d=0.5, you see for 10 layers, Rac/Rdc<2. You'd probably want to use this graph to match the wire thickness to the number of layers for Rac/Rdc<2or3
Yes, amps RMS are what's important since we're trying to calculate power dissipation. This would generally be significantly less than the peak current.