Why isn't there a category for New Game+? If you beat the game once and then look at your save file it should say new game and be brown, from there you can beat the whole game in Link's Pajamas :P

Mostly because there isn't any animo for it. I'm not part of the mod team, but I'm sure that it might get added if there is a calling for it.

The differences in NG+ compared to a normal file making speedrunning slower for all categories of the game (except 100%, but it's banned in 100% because it starts you with the deluxe pictobox) so there isn't really a point in doing any runs with NG+ for most categories if you want a good time. There are some people who will occasionally do 100% NG+ All Figurines runs for fun though. If you do runs in NG+ you can still submit them for any category besides 100%, but you'll be at a disadvantage compared to the rest of those who run on normal files.