You mean 5600X? There is a stock cooler, it's good enough unless overclock, and I didn't really hear the fan in my case, but full metal case.
You can enable AMP (XMP for AMD) on the board and let the RAM clock higher, or just leave it at 3200MHz, it will have tighter timings then, but 3600 will be fine, you can also set it yourself, not a high clock so should be fine. It's not like Intel where they lock the RAM speed. 3600 is about a 5% increase in most titles if same timings.
Feel free to swap the mobo, there was no stock of it at the time and I know the Aorus board is good in terms of VRMs.
I just find 16GB of RAM cutting it for a lot of games nowadays if you play things like Space Engineers, or leave other things open while gaming, think Mankind Divided pushed 12GB, but can't remember anymore, but again, you're just playing BF, so feel free to do later. I would not go under 16, there are enough titles now that can't go under 8GB.
If you want to go 5800X cooler, I would get:
https://www.amazon.com/-/en/Noctua-NH-U12S-Ultra-Quiet-Cooler-NF-F12/dp/B00C9EYVGY/
Just because Noctua so far has always just sold mounting kit separately so can carry to next generation, so you can just buy the mounting kits to a new socket and keep taking the cooler with, e.g.
https://noctua.at/en/nm-am4-mounting-kit and then you just pay shipping. It is considered one of the best coolers on the market (if not the best) with best noise/performance.
I don't think the upgrade is really worth for gaming unless you intend to stream, do work or video editing or something.