1) 2d platformers: impossible for me to say, 2d platformers are probably my overall favorite genre, the early days (8 and 16 bit systems) were their glory days but from xbox 360 to today they have seen a major resurgence and are just as strong, if not stronger, then they ever were.
2) run n gun: probably have to go with 16 bit era here(includes neo geo), as much as I hate to forsake contra and super c, we had gunstar heroes, hard corps, and contra 3 during this time frame and the top down shooters in the 16 bit era were much stronger than any other time frame
3) FPS: probably xbox 360 to today, I generally prefer the more modern style of fps to the old style, although I have a special place in my heart for OG doom, I would take a modern FPS over an old one just about anytime.
4) Rail Shooter: this is an easy one, wii generation hands down, not only do wii light gun shooters work on modern tv's but the quantity and quality of these games are vastly superior to any other console generation, not even close. You could make a case for the arcade scene during a certain time frame but I'll stick with the generation that gave us house of the dead 2-4, overkill, dead space extraction, resident evil chronicles, and so much more
5) 2d beat em up: no brainer here, 16 bit era is king, both in the home console scene and the arcade. If you include openbor games the water becomes much more murky as I have played many open bor games that I enjoyed even more than streets of rage 2 and turtles in time, but if we are strictly talking commercial releases than its a no contest.
6) Rhythm: PS1 is the winner here, I am not a huge fan of this genre but parappa and bust a groove are both epic games for me and I still often listen to songs from bust a groove when I'm lifting, just this morning I was listening to heat and gas-o's extended tracks while squatting so those games are timeless for me.
7) 3d action and advenutre: this is a pretty broad category, if you include stuff like god of war, devil may cry, bayonetta here than the answer is modern years (gen 7 to today)is an easy one, this is a genre that just gets progressively better