The RX 580 is a good GPU for 1080p gaming. It's still good for the money, especially since some go for under $200 nowadays. The GTX 1660 Super seems to be it's closest competition, which does perform better on average but is a bit more expensive.
Those are called 6+2pin pcie power connectors, they can be used in either an 8pin or 6pin power connector on a GPU.
It Depends on the GPU, the model, and vendor of said GPU. I'd say most cards in the RX 580s price range have a single 8pin. My 2070 super for example has an 8pin and a 6pin.
I wouldn't go from a 580 to a 1660 super, not worth the money imo. I'd wait for the next round of GPU's from AMD and Nvidia before upgrading from the 580 if I were you.
Yes and no again depends on the GPU and the power requirements of the rest of your components and whether or not you will be overclocking. I run a 2070 super with a 550watt PSU (which is not necessarily recommended) however it is a quality Seasonic G series unit and im yet to have an issue so im not to worried about it. If I had some off brand low quality 550watt psu I wouldn't even consider trying to run a 2070 super with it.