[PC Build] Need some advice choosing CPU

9800X3D is the best gaming CPU currently, that is a good enough reason I think. :)

Both Intel and AMD would be good with this GPU. Remember to choose efficient heating system.

If you got the budget than 9800x3d sure, but otherwise something like ryzen 7600 or even 7500f is an affordable option. Definitely would not go with Intel these days. AMD tends to honour their socket for 5-7 years and AM5 is relatively new. Intel on the other hand switches their socket every year or so.

I would definitely go for Ryzen. I hope it's useful. Cheers.

quite frankly depends on your needs.. But as you didn't specify I would say 9800X3D and don't forget to buy 2 ram sticks(for dual channel) instead of single dimm and also don't cheap out and go for 32gb (16x2) total at least, 64gb (32x2) for your budget is good, don't really need more for now (for gaming). (Don't listen to those 16gb is enough as they I guess min max stuff to make it work or just haven't used more. I don't have a clue why someone says that...)

The 9800X3D is the best gaming CPU but I am not sure if that is the main purpose of the rig. Depending on the resolution and games you will play, the 9800X3D can make a small or big difference. But it does not have the best price/perf. on the market, there are better options available.
I have also no clue what RAM and MB you want to pair it with. Technically everything above DDR5 5600 is OC and not guaranteed to be 100% stable. That said, 6200/6400 MTs RAM should not a problem but from my experience the AMD CPUs are more picky with RAM and motherboards. Currently I would avoid Asrock mainboards, because the chance of having an issue is higher (even with the new BIOS that is supposedly fixing it).

tl;dr 9800X3D is the best gaming CPU, but pricey. If you have less budget left or want to use the PC also for work there are other options available.
Edit: If you want the rig just for gaming, you don't need more than 32 GB. Rather save the money and get a better graphics card or CPU.

From your choice of GPU (RTX 5080) I guess you're intending to play in 4K or 1440p at least. The higher the resolution you play the less important is your CPU. Those Intel CPUs get very hot as they are power inefficient so you'll need good cooling. If you don't mind price my choice would be AMD 9800 X3D as this CPU will be still a viable option with a 6090 or 7090.

I'm playing in 4K on a RTX 4090 and 7800 X3D so I know what I'm talking about.
https://i.imgur.com/w70wqw6.jpg

9800X3D, easily. The extra cache memory on X3D CPU's is incredibly useful for gaming performance.
Intel has nothing atm that can compete with it.