When throttle is applied, right wheel drives, left wheel locks within a few feet, keeps moving, but drags left wheel.....

tune up and or carb overhaul indicated

I assume that this is a hydraulic drive. These usually have an isolating rod which has to be pushed in or pulled out, depending on model, before manually shifting machine. This opens a bypass valve which makes the drive ineffective. Check that this rod, which will be low on the rear of the machine, often on the left hand side, has not been moved. Try moving it and see if this is the difference.

to push start a motor bike you have to use a higher gear like third gear but if you are only pushing it from one place to another then it is better in neutral or out of gear

I believe that is a safety for if your on an incline you do not just free wheel uncontrollably. But that is not a 100% answer

yes replace

yes this is correct that is called a " differential" it allows you to turn in circles without one wheel dragging in grass, larger 4wd drive tractors have what is called a "diff Lock" that can be engauged this pins both tires on same axle together and keeps them both turning in low traction surfaces,

im unaware of any craftsman that is made with a "diff LOck"

Jack the car up put you hand on the 12 and 6 o'clock position and try to rock the tire if it moves it is a bad wheel baring if it squeaks when applying the brakes it can be the pads are worn out if it clicks I can be the half drive shaft if it is a drive shaft run one way if it is quit it is ithe inside it will be noisy if you make circles the other direction then there are the strut towers that can squeak hope this helps

you mean parking brake cable? loosen parking brake cables put drums on , and adjust shoes till they lightly drag, now apply brakes a few times and check the adjustment again, if all ok, now adjust park brake cables, you want to adjust so park pedal goes 5-8 clicks till the wheels are "locked up" and the wheels spin free with the park brake off

hello , it sounds like maybe you need to have your car alimement check .