New answers tagged remote-control
Google's Android Device Manager service can do all of that apart from the remote control, though unless your phone has a new enough OS to have it installed by default (instead of pulled down as part of the Google Services updates) it's difficult to remove rather than impossible. Find your lost phone with Android Device manager Remotely lock Android devices ...
Making an app "impossible to remove" is, of course, impossible to accomplish. Nothing is impossible to remove. For other features you're looking for, I'd recommend looking into AirDroid or Cerberus.
Top 50 recent answers are included