The type of freewheel or cassette is totally irrelevent to whether the wheel sits properly in the frame. If the rim is properly dished and the frame/dropouts properly aligned the rim will be centered. It is true that the rim may need to be dished further in respect to the hub flanges with some freewheels/cassettes, but it is always centered over the locknuts. Either the wheel is not properly dished or the frame/dropouts are not properly aligned. If you bought the wheel from a shop ask them to check the dish. If not just reverse the wheel in the frame. If off in the opposite direction respective to the frame) the dish is wrong, if in the same direction the frame has a problem. Also note that it is very common to have to jockey the wheel a bit to center it on less expensive bikes.

Dishing is not just moving the rim over but also keeping the tension appropriate, and moving washers is dependent on what clearance you have on the right side for the chain as well as your derailleur adjustment limitations. In the old days we used to move spacers when we could get away with it to minimize the amount of dish or to accommodate a change in freewheel or rims when things were not very standardized. Do not do so without understanding the ramifications.

Can't answer your questions fully until you determine if the wheel is the problem and in which direction it is off center.

"Respacing the axel (sic) to be centered on the frame" is not appropriate - not sure what that means.