Yes, your washing machine should be level and all legs should be squarely on the floor. If it's not, you'll have unbalanced forces acting on it as the drum spins. During the spin cycle, these forces are much larger because of the higher spin speeds and will make it "walk".
You'll need a spirit level to balance the washer. There's usually a gap between the top plate and the side plate of the washer that you can rest the level in to measure the front-to-back tilt. For the side-to-side tilt, generally just put the level on the top of the washer. You may have lock-nuts on the legs: if so, loosen them with a wrench before trying to adjust the washer, and tighten them afterwards. Raise or lower the legs until all four are solidly on the floor; the washer should not rock at all. If you have to move the washer to adjust it, remember to check it once it's back in its place.
It is normal for resistance to be felt when attempting to rotate the motor driveline attached to the gearbox. Gearboxes have a fair bit of friction and are attached to a rather heavy tub. This machine is equipped with a tub brake that is supposed to engage during the spin cycle if the lid is lifted. The spinning stopping fairly quickly is normal. Continuing to spin indicates the brake has malfunctioned. I'm not sure if you are referring to the wash or spin cycle though. If it was wash cycle, it does sound odd, but I doubt the coupler would be at risk.
When off, the inner tub should be difficult to turn, but it should turn, you are turning the entire drivetrain and motor rotor. It cannot be deduced that there is a problem with the motor or gearbox with the information given, but I think failure of either is unlikely. I'm actually not hearing anything indicating anything is seriously wrong at all.
Since you asked, a new gearbox mail ordered in the US has a list price around $200 plus shipping. I would expect a rebuilt one to be 2/3 to 3/4 of that. Replacement is disassembly of the whole tub/agitator assembly, a fair bit of work.
Interestingly, another person is having difficulty with his Kenmore machine not spinning. It's well known that Kenmore appliances are rebadged name brand appliances. The parts diagrams of his and your machine look remarkably similar, if not identical. Probably an unrelated problem though, it's just curious.
Best Answer
Motors don't break... the brushes used to power the coils or the golden fingers used to get the power are for sure worn out and need replacing. This can be a difficult task sometimes and these parts could cost $50, so by replacing the motor with an equivalent type you should expect the same life span based on the usage of the machine.
In this picture you can see the brush (inside the yellowish bracket) and the commutator (golden fingers) it touches (here they are dirty - but in good condition) usually they would need to get cleaned on change of brush - but if they are badly worn the motor could be rendered useless as you can see - they are not replaceable on this version.
A nice picture showing the complete design of a coil-motor. The only things that break in here are the brushes/commutator and in rare cases the coils themselves could burn if a high current flows past them. The coils never burn from just running. These motors can run forever as long as they have power to them (even if you physically stop the shaft and put 100% power they won't burn); it just generates strong electromagnetic fields.