Are there legal limits associated with the use of the autopilot in the US?
This varies based on if you are VFR or IFR certified. 

Basic currency requirements for general aviation flying under part 91 which I presume is the target of this question. The comments to the answer outline the commercial implications, can be found in 


  §61.57   Recent flight experience: Pilot in command.
  
  (a) General experience. (1) Except as provided in paragraph (e) of
  this section, no person may act as a pilot in command of an aircraft
  carrying passengers or of an aircraft certificated for more than one
  pilot flight crewmember unless that person has made at least three
  takeoffs and three landings within the preceding 90 days, and—
  
  (i) The person acted as the sole manipulator of the flight controls;
  and
  
  (ii) The required takeoffs and landings were performed in an aircraft
  of the same category, class, and type (if a type rating is required),
  and, if the aircraft to be flown is an airplane with a tailwheel, the
  takeoffs and landings must have been made to a full stop in an
  airplane with a tailwheel.
  
  .... [read on for IFR and night currency]


However the FAA actually views using the auto-pilot as manipulating the controls. You can find the full legal interpretation here. But in short, 


  The FAA considers a pilot's use and management of the autopilot to be
  the equivalent of manipulating the controls, just as one manages other
  flight control systems, such as trim or a yaw dampener. The autopilot
  system's sophistication does not affect a pilot's responsibility to
  manipulate and manage all control systems, including an autopilot,
  appropriately. Therefore, a pilot may log PIC flight time as the sole
  manipulator of the controls for the time in which he or she engages an
  autopilot.


With that in mind you can maintain your currency using only auto-pilot and in theory if your plane and the desired airport was equipped for auto-land (which, as far as I know, no GA planes are) you could maintain your currency without physically manipulating the yoke aside from takeoff and climb to autopilot altitude. So no there is no legal obligation to physically touch the controls to maintain currency if your plane is capable of some kind of completely programmable flight (I don't think any are currently).  

There is however one notable exception to this regulation. On an IFR checkride you must fly a manual approach, 


  From the ACS...
  
  • The applicant must accomplish at least two nonprecision approaches in simulated or actual weather conditions.
  
  
  At least one must be flown without the use of autopilot and without
  the assistance of radar vectors. The yaw damper and flight director
  are not considered parts of the autopilot for purposes of this Task.
  





  It claims to do everything for you within 50 feet of the ground.


This is a bit misleading. First off, most, if not all GA planes lack auto-throttle so power management is in the hands of the pilot and is fully manual in most cases. Some more modern airplanes like the DA-42 have FADEC which loops prop and throttle controls to one lever minimizing but not eliminating manual control. 

There are lots of autopilots out there and function varies heavily: some only have a single axis (heading hold and adjust capability). Some have 2 axes which can hold altitude and headings and some which have the ability to follow GPS courses, fly approaches down to different minimums etc. It should be noted that some units can be set to create situations that can stall the aircraft or otherwise put it in a dangerous flight scenario, the auto in autopilot does not mean absolute protection or full automation.  




  If so, can it be remotely piloted?


This is actually a completely separate question if you are asking "do I need to physically be inside the aircraft" i.e. "can I maintain currency with a drone if it's properly equipped". I don't think the FAA has fully fledged regulations on this yet. If by remote you mean by use of autopilot, see above.