I haven’t actually had a chance to play with a KK bearing yet but I understand it keeps the string easily on the center since the exterior part of it is grooved. Are there any downsides to that? Why doesn’t every new yoyo come with a KK bearing?
If I am not mistaken the concave bearing is patented by difeyo. YYF just came out with a version very similar but different enough to not infringe on the patent.
I enjoy using concaves in most of the yoyos I play with the most. It is just a different fell for the most part. They play differently then a normal bearing. Nether is better then the other. Its just a personal preference.
No they fall the same as any other bearing. Pretty simple knowledge there. 
But yes, it’s a different feel. Some people like it, some people don’t. Also, good players sometimes don’t use them because a good throw will center the string by itself usually, so there is no need for it.
I’ve never understood why some people do not like konkave. I understand that some people might not “need” them, but what’s not to like?
The only downside I can think of is that you can be a bit more sloppy with your play. But I don’t think it “feels” any differerent, and it’s not like a konkave bearing will let you go auto pilot and every throw will be smooth. You still have to make some effort into making good throws 
I got a KK (for YYJ and most metals) and a Wintergreen Peak yesterday. Will the KK fit into the Peak or am I gonna end up using it in my GM2?
If you are really experienced, you do notice a difference between a regular bearing and a Koncave.
Your Peak will have a C sized bearing, the same as a Koncave size (the large one that is)