Yes, the Wii contained full GameCube components inside, so it was able to natively play GameCube games. I think the later Wii Mini model removed that capability, but the regular Wii model was backwards compatible.
Yes, ours had that. There was a door on the top that concealed four GameCube controller ports and two memory slots. We never had a GameCube but I ended up buying some controllers and GC games to play on the Wii.
There was a model before the Wii Mini that also dropped GameCube support. It looks similar to the original model but lacks the GameCube ports and the "Wii" logo is placed for horizontal orientation instead of vertical. It's sometimes called the "family edition".
The Wii Mini originally released in Canada but came to other regions later. Japan never got the GameCube-less models for some reason.
The article mentions that the Wii U had gamecube compatibility. I wonder how that was achieved? And I'm guessing it can't take the small optical disks of the gamecube.
The Wii plays GameCube games and accepts the controllers and memory cards, the Wii U doesn't. I assume removing the ports and the mechanism to load small discs in the disc slot were the main reasons for dropping it.
The Wii was compatible with the GameCube largely by being basically an enhanced GameCube under the hood, and the Wii U is kind of a further step down that same road with its "vWii" backwards compatibility mode, so it's possible to use homebrew to play dumped GameCube games on a Wii U natively, only providing some light glue to hook things together. It's not officially supported though.
The Wii U could play gamecube games in wii mode, but it required jailbreaking and loading the images onto storage. So, I wouldn't say it was "possible" ootb.