Is it worth upgrading to a Ryzen 5 1600 if I'm already running an i5 6600K?

No. If all you're doing is dicking around on the internet and playing PC games you're better off where you are. You might get a slight edge with the Ryzen in heavily threaded stuff, but since you'll have to get a brand new mobo to go along with it, you'll never see the value in spending that kind of money.

As for the RX580, forget about it for now. Thanks to the recent uptick in Ethereum mining most of the mid-range graphics cards are sold out or getting severely marked up. Wait a few months for the market to correct itself before buying a graphics card.

If you're just gaming, that would actually be a bit of a downgrade. If you're going to make videos or livestream, then it would be worth it. And as for the GPU, the 580 is faster than the 1060 by about 1-2 FPS at most depending on the game. And don't crossfire. Always get the single more powerful card.

No, it's not.

Nope to the 1600x

Nope to Crossfire or SLI. It's not viable anymore except in a few older games. No one recommends it anymore and there's a reason why.

I don't get your question at all because you're talking about sidegrading to AMD, not upgrading. The RX 580 is like comparing apples to oranges with a 6gb GTX 1060. People spout off that one is a few percentage points faster than the other but 5% or 10% doesn't count when your overall number is 55 or 60. The Ryzen processors have cache latency limitations which puts it at a disadvantage against 4th generation and up Core i5 and Core i7 processors clocked beyond 4.2ghz.

For Overclocking........ go read one of these online guides you can find through Google. There are hundreds of guides and information in old forums. Overclocking is relatively harmless if you are prudent with your voltages, have good cooling, and double check for actual voltage usage and stability.

All RX 580, RX 570, GTX 1060, and GTX 1070 cards are sold out so now in not a good time to buy one. The only viable upgrade you can make in this present moment is to move to the Core i7-7700k. This wouldn't give you a huge leap in speeds or anything but you could sell off your 6600k for $150 and pay for half of it. You would need top of the line cooling for the 7700k too because it's such a hot processor.

since you'd need a new mobo, i'd not do it,

and the rx 580s are all sold out, so that's on delay, too