Are there any speed and altitude limitations on the autopilot?
It depends on the autopilot and you need to read the AFM to know what the limitations are. 

All of the business jets that I have flown prohibit the use of the autopilot below a specific altitude (between 50 and 1,000 feet depending on the autopilot and the configuration of the airplane). Nothing prevents then from being used lower but it isn't allowed.  They will engage on the ground (in fact it is part of the required preflight tests) but have a specific prohibition against use for takeoff and landing. 

Other airplanes and autopilots however are approved for autoland all of the way down to a full stop, and some for takeoff as well. 

Speeds also depend on the autopilot.  Some automatically disconnect if the aircraft receives a stall warning or overspeed signal but others don't.