First of all, most of the info I'm giving you here, you can also find over at the
smashladder controller guide.
This won't work with dolphin, at least not the way you see in all the guides. There are only few controller adapter that support native controller inputs, meaning they pass on the inputs directly. The instructions you saw with using Zadig are only for these kind of adapters. Specifically, those adapters are the official Nintendo GameCube adapters and the mayflash 4 port and mayflash 2 port adapter with a switch on the back. You have to set the mayflash adapter to "Wii U" mode with the switch. I strongly recommend getting one of those if you play competitively as all generic alternatives have input lag and the control stick isn't accurate.
Generic controller adapters like yours work by creating a virtual gamepad in windows. Making them work in dolphin is quite easy, you can probably just look that up. First make sure your controller shows up as a gamepad in the windows gamepad settings (type gamepad in the windows search bar). If it doesn't show up, visit the manufacturer website and download drivers or firmware updates for the device.
Setting a controller like this up in dolphin is easy. Choose "standard adapter", then choose "configure". Then you just tell dolphin which button is which in the setup window, it's quite intuitive. Look up a YouTube tutorial if you get stuck.
