Nintendo Switch – Is It Backwards Compatible?

No.

From the Nintendo Switch FAQ:

What controllers work with Nintendo Switch?

There are multiple controller options for the Nintendo Switch console, including Joy-Con controllers and the Nintendo Switch Pro Controller. Compatible controllers will vary per game, so be sure to check game details. You can see available accessories here.

Controllers from previous Nintendo platforms are not supported.

(emphasis mine)

As noted above, the Switch has its own Pro Controller. It's more expensive than the WiiU Pro Controller, likely because it has motion controls and a Near Field Communication (NFC) reader (for Amiibo) built in.

Yes, but only if you have an Amiibo to transfer it (which it sounds like you already figured out).

Nintendo answered some questions Kotaku had. This was in one of the responses.

You can send/receive Mii characters through local wireless communication between Nintendo Switch systems, or use an amiibo figure to transfer Mii Characters from Wii U and Nintendo 3DS systems.

Having said that, it sounds like this is a uni-directional transfer as Switch Miis have some new properties that previous Miis did not have.