Quickly responding to a few points here:

AirPlay is WiFi/Ethernet only, not bluetooth. Some confusion exists here because the airplay icon is also used to change the sound only input to a bluetooth audio device. Full video based airplay, or the audio multiple speaker airplay is IP networking only. It uses bonjour, so if the network in question is not bonjour friendly, it's not going to work.

Yes, Airplay and Keynote work fine, and the iPad will either do direct mirroring, or output a different video feed (in this case the presentation) while showing something else on the iPad.

Paring a Bluetooth mouse to an iOS device does nothing. The OS has no concept of a mouse, mouse cursor, or mouse clicks.

Really, the proper solution here appears to either just use a laptop, and the iOS Keynote Remote app to drive it as a controller. Or use the iPad directly, using an AppleTV for wireless, or using the VGA/HDMI dongles.

And as far as 1080p, it will come in time. 720p was picked for a number of reasons. Yes, the A4 can mostly handle 1080p. Add in some overhead though (such as processing an incoming AirPlay stream while possibly downloading an OS update in the background), and the experience may fall below the thresholds Apple finds acceptable.