As far as number of cores, it is totally dependent on the applications you will be running. If the applications do not take advantage of multi-core processing, the extra cores will be of little value.
Yes. A fast (high clock speed), 4 or 6 core will suit most needs, including gaming, so I'm told. (Games use a few cores intensely, but not so "multi-threaded"?)
I've read on the Ryzen 5, 5600X... a 6-core with a Passmark score of ~23,000 and a single thread rating of 3380. That's fast! Not expensive either... $350-$400-ish)