Can you download Fortnite on the iPad (if so, can the controller be connected to the iPad somehow)?
Yes you can connect a Ps4 controller there.
Only ps4?
I'm not sure, but there are certainly other controllers that fit.
https://www.playstation.com/...os-iphone/
Further information is possible here.
Edit:
The following wireless controllers are supported
Xbox Wireless Controller with Bluetooth (Model 1708)
PlayStation DUALSHOCK®4 wireless controller
Bluetooth controllers labeled "MFi" (Made for iOS) such as SteelSeries Nimbus, Horipad Ultimate and others may be supported.
According to Apple.