On the CPU cooling, I have two different ones running (both currently in the open without any case). Maybe it helps someone.
Noctua NH-U12S: Covered by others and a pretty safe bet. As long as you get the version with the Intel mounts. I heard others unglued the mounting plate to fit it with AM4 parts, but I went the other route:
Get 4 m3 screws with nuts and screw them in through the backplate from underneath. The Noctua plastic spacers can then be fastened on those and the rest of the mounting is according to the manual. There were three different standoff sets with very slightly different height and I was not sure if the AM4 or Intel ones were the best fit but went with the back ones.
This cooler was quite a nerve-wrecking ordeal to fasten for me (I don't think that part is particular for the X570D4I, though), but once put on it sits snug. RAM and PCIe slots are really close, but all are clear without unmounting. I dread the day I have to remove this cooler. CPU is cool and happy without the fan spinning up notably, but other components run hot and I need to point a separate fan at it even without a case.
Scythe Choten aka SCCT-1000: Comes with stupid snap-on feet that have to be removed, and likely to break in the process. I used the same strategy as for the NH-U12S (m3 screws through the board). The distance after that is way off, though - the Noctua standoffs are several mm too tall and without is too short. Maybe one can replace the brackets with other ones. Instead I got some spongy M4 foam spacers from the hardware store that I cut to size, threaded onto the M3 screws, and sandwiched between plastic M4 spacers and the cooler brackets, fastening it with bolts. Also a bit of an anxious installation since it's hard to tell if you tighten it too much or too little.
I just installed it yesterday and it seems to work fine. Since it's a top-down cooler, it also blows down air on the other hot-running parts, like the X570 chipset. Plenty of clearance for RAM. A full-size PCIe card will just about fit, but only if there are no components or heatsinks on the side facing the cooler.
My testing hasn't been rigorous yet, but so far the SCCT-1000 seems like a much better choice for this mobo than the NH-U12S - it's significantly cheaper, takes up less volume, isn't notably noisier (if at all?) under load and gets airflow for other components as well.
I read in comments that the Noctua DH-L9i should fit as well, but at that price I didn't want to take any chances since I'm not clear on if it blocks the closes RAM slot or not.