Cannot fit wheel into dropouts
You need enough spacers behind the cassette so that when you tighten the cassette lockring there is no play of the cassette on the freehub. Probably 0 on 10 speed freehub, at least 1mm for 10s on 11s FH. The cassette shouldn't touch the frame in either case, so having a spacer shouldn't have prevented you installing the wheel or cause misalignment. If it was the cassette touching the frame that stopped you mounting either wheel there might be scratches to evidence this but it seems unlikely. 

As commented, check that the wheel fits without the cassette fitted at all. Also check that the axle locknuts are tight (i.e. that you can't unscrew anything from the axle with your fingers). Check that there is an even length of threaded axle exposed on both ends (roughly 4-5 mm). 

The most likely suspects in my experience, however, are the springs on the quick release skewer. Presumably you have had to remove the skewer to change cassette spacers. If you put either of these on backwards then they can obstruct the axle entering the dropouts at all, and will cause the wheel to be out of alignment if it does go in, maybe causing tyre rubbing on the chain stay. The springs have a narrow end and a wide end and the wide end goes on the outside on each side. The spring never sits over the threaded axle. There's a picture on http://bicycletutor.com/remove-install-wheels/ 
If you were doing this correctly all along then I'm all out but this might help someone else