Can the original white Wii remotes (with the add on motion plus adapters) be used to play games on the Wii U?
Yes, unless the game only lets you use the Wii U Gamepad. Check each game, though most have an option to play with a Wii Remote.
Can I play the newer Mario Kart 8 with the old original white Wii remotes and steering wheel or do I actually need to go buy new Wii U remotes?
No, you can use any Wii Remote on Mario Kart 8.
I plan on keeping the old Wii system and put it in the bedroom to watch Netflix, but wanted to know if I HAD to buy newer Wii U remotes when I already have 3 functioning ones from the older system.
Not necessary. The new Wii U ones are the same as Wii Motion Plus/Wii Remote Plus controllers, just with new branding. But even then, original Wii remotes without Motion Plus will be fine for most things, including navigating menus. You only need Motion Plus for games that take advantage of it like Skyward Sword or Wii Sports Resort.
