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Yeah, the cheapest one you can get will be more than enough. They can even handle an overclocked 8700K just fine.
Pretty much anything will be better than the stock cooler. Which isn't to say that it can't do the job, but from people I've seen talking about the 8400 with the stock cooler, it can get pretty noisy under load. At the very least a third party cooler will be quieter.Do you think something like this would be a bit better than the stock Intel cooler? https://www.overclockers.co.uk/cryorig-c7-top-flow-cpu-heatsink-hs-009-cy.html
Any board will be good, but obviously make sure it has certain features you may want. I made a build with the bottom tie asrock z370 A and it was fine.
The stock cooler is noisy and crap. On my 8700 (non k) build i have just completed, in realbench temps quickly went to over 80 degrees and it was very noisy. Installed a Raijintek leto and now it seems to max out around 60 in realbench and it is silent. Really pleased with the leto actually, very nice looking , decent cooling and very quiet. Not one if you are going to be doing extreme clocking but great if you are staying at stock.
For my 8700 build i went for a gigabyte gaming k3 which works out very cheap if you get a good price on it and sell the steam code you get with it.