Same hub? Doesn't sound like it. Gears slipping sounds like a stretched chain & worn teeth. If the wheel is clamped in with a quality hub & skewers, it shouldn't be moving in the dropouts.
The hub width shouldn't change when lacing up a new rim. Most common hub widths are in 1/2 cm increments 126, 130, 135. Thus tells me something went really screwy with your hub, or more likely the just grabbed another wheel or laced it up to a different hub. That's my guess