I have an Edge 25.
You cannot transfer rides/courses over WIFI. You can upload courses to and download rides from the Edge 25 via Bluetooth from a smartphone.
I don't know why you can't pair the Edge 25 to your iPad via Bluetooth and can't imagine why it wouldn't be supported. I have it paired with an iPhone 6 running IOS 8.4. There were a lot of problems pairing with 8.1 so you may want to upgrade your iPad if not already. The only other suggestion I have is to make sure the Edge 25 is updated to the latest firmware, version 3.0. Use Garmin Express to update it if it is not already 3.0.
If you do get it to pair with an iPad then you are good to go. Just download the mobile app as Seattle Forest said above and you can sync all you want.
Mine is version 3.0 firmware and Bluetooth can be turned on/off under Settings --> Bluetooth - Turn on/off.