I'm planning on changing my pc case to Nzxt H500 and I'm not sure whether I have to remove the CPU AMD Ryzen 5 1600 with its stock cooler from the MB. And also whether the 2 stock case fans are enough. I'm not planning to OC anything
I haven't had chance to work with a Ryzen system yet myself however if the stock cooler is designed like ones of old I would assume you can simply disconnect all cables to the mobo and lift it out, although be careful if the heatsink is quite heavy/large as it could put some stress on the board when lifting it out (providing you hold it from each corner and the CPU is more central). And on to your other question about the stock case fans the only real way to find out would be to fit it inside the new case and run some sort of benchmark or game heavily on it and monitor your system temps using a program like Speedfan.
EDIT: Just looked up the stock cooler, should be absolutely fine.