1) NVidia has better DX11 drivers, so any advantage there is solely down to whether you are experiencing a CPU bottleneck or not. If YES you should get a higher FPS but it's pretty hard to compare with two different GPU's.
I don't think this is significant with your CPU and any losses there in some games should be under 10% and as I'll discuss OTHER games will see an advantage to the AMD card anyway due to architectural differences.
2) AMD CPU itself does not impart an advantage to an AMD graphics card.
3) asynchronous monitors are tied to the GPU (so Freesync for AMD and GSync for NVidia).
4) GTX1060 6GB vs RX-580 8GB?
*I'd ignore points #1 and 2 since they are mostly not an issue and look at price vs performance for cards of similar build quality. Having said that you'll want a BETTER cooler for the RX-580 or the noise will be higher as it produces more heat.
Example GTX1060 6GB:
https://pcpartpicker.com/product/MDRFf7/evga-geforce-gtx-1060-6gb-6gb-acx-30-video-card-06g-p4-6262-kr
$275USD (after rebate) for a dual-fan card
Example RX-580 8GB:
https://pcpartpicker.com/product/9ZmxFT/asus-radeon-rx-580-8gb-rog-strix-video-card-rog-strix-rx580-o8g-gaming
$300USD
Prices change but if I was forced to choose I'd get the Asus Strix RX-580 8GB card... if you get a lower-end solution the quality may not be as good and the cooler may be insufficient to prevent throttling.
The RX-400/500 cards have some mostly untapped potential that helps the pull ahead a bit for newer games that optimize for it (mostly stuff like FP16) so you might gain 15% or so due to that.
Generally speaking I'd say these cards are close enough in performance to be called roughly EQUAL.
Summary:
Honestly I think both solutions will give similar results provided the card has a relatively good cooler and is of good quality.
If I was forced to choose it would be specifically the RX-580 8GB Asus Strix model if the price is about $300USD. Otherwise adjust my comments based on price.