Who is responsible to check for radar-altimeter disagreement?
That depends on regulations but broadly speaking it is ultimately always the Pilot In Commands responsibility to make sure all instruments are functioning properly. 


  § 91.3 Responsibility and authority of the pilot in command. 
  
  (a) The
  pilot in command of an aircraft is directly responsible for, and is
  the final authority as to, the operation of that aircraft.


Now your probobly thinking "But if the autopilot is flying then who's really in command" well its the opinion of the FAA that even if the autopilot is taking care of everything you are still responsible for the aircraft and for making sure everything is working correctly. This question covers that more in depth.

As for things like warnings, alerts and auto-disconnects that depends on the specific system in question and the desired failure mode. For example if you have 3 radar altimeter inputs to an autopilot/auto-throttle the system can reasonably detect if one is erroneous and use the other two. There is lots out there on triple redundancy which is the basis of this. 

What if the aircraft starts barking alarms at you saying "TOO LOW!, GEAR!" how can you even check that aside from cross checking your altimeters? 


You can check your GPS altitude if available, while not a legal source for navigation you bet its a good backup. 
If you are flying a stabilized approach (and in a plane like that you really should be) you should, at any given time know generally where you are on the approach and if you are on the glide slope (when available) you can gain a pretty good idea of your altitude. 
In VMC, just look out the window!