Do regular Wiimotes work on the Wii-U?

Yes, regular Wiimotes are supported by the Wii-U.

This page on Nintendo's website says:

Controllers

The Wii U console is capable of supporting two Wii U GamePad controllers, up to four Wii Remote (or Wii Remote Plus) controllers or Wii U Pro Controllers, and Wii accessories such as the Nunchuk, Classic Controller and Wii Balance Board.

This article from Eurogamer seems to suggest that it doesn't.

Nintendo's Wii U console will not support original models of the Wii controller without a MotionPlus attachment, according to comments made by Shigeru Miyamoto.

Yes, you can but only on games like Mario. Call of Duty will need the motion plus wiimote unless you are in the wii software on your wiiu. If you did not know your wii is two different consoles with a channel to switch back and forth between them.