Controlling Mothers Ipad

iOS/iPadOS has no facility for remote control or remote management of the device. There are no third-party Apps that offer this facility.

The best that you can likely achieve is to use FaceTime (or other communication platform) - and literally “talk-though” a problem; if you have similar hardware, and the same version of iPadOS installed, you should be able to work through a problem step-by-step.  iPadOS15 now allows you to “share” the iPad screen; this new feature may help with resolving issues remotely.

Absolutely - but both yours and her iPad must be running iPadOS15 or later.

See >>> https://www.macrumors.com/how-to/share-screen-facetime-call-ios/

Yes. Share Play / Screen Sharing can be used over the internet. However, it is not yet available on iOS 15, it will come in an update later this Fall.

Also, your mother will need to be able to initiate screen sharing herself, you cannot start it for her on her iPad remotely.

Yes - as you’ll be sharing the screen over a FaceTime call.

As FaceTime calls are between users that may be separated over considerable distances, you both need to have either a WiFi or Cellular connection. You do need (indeed are very unlikely to be) to be using the same WiFi network.