You mean something like this ? https://itunes.apple.com/us/app/remote-mouse-free-wireless-k...

Accurately inferring the position of the mouse from its acceleration is a difficult problem (the integration involved will result in rather large errors that compound over time when calculating the velocity and position). I doubt that the return on a 'traditional mouse' app would reward the effort required to make this work.

I don'f quite understand what you want ot build. In general a moise only makes sense for screens bigget than a typical mouse-pad or phone screen or look arround stuff like 3D games

I thought about this a long time ago (2009) using the gyros + compass as an IMU. The thing is that the gyros were not as precise back then but things might have changed now.

Probably not too much. Mice are cheap and everyone has one, I don't think there is much market for a phone-based one.

I already pair a bluetooth mouse to my phone...