Front end shake

It sounds like a couple issues. The shake at speed could be tire related if it started right after the tires were installed, not uncommon for a weight to get tossed or slip sometimes. Were the wheels installed at the same time? If so, then I would check the balance and for tire runout as well. Some rims mount in a way that can make them act this way as well, they are the types that don't center on the axle hub and instead use the lug nuts to center, they can be touchy if they are not installed just right.
The side to side on a bump could be related to them installing the tires, if you have worn suspension bushings, raising the car during the tire install could have shifted them or even caused them to come apart so things can now shift. If you can get under the car, you could have someone push down on the car and watch the circled areas for movement other than just up and down rotation. I would suspect the track bar bushings more (in orange) but they all could do that. The motion could also be normal as the track bar will try to push the rear end sideways as it moves, but it shouldn't be really noticeable.

Track bar bushings are the items I circled in the picture. It holds the rear axle in place, so it doesn't move side to side. However, it sounds like you have an issue with it if one tire is sticking out. Is this car lowered or stock? If stock, there is an issue with either the track bar or the mounts for it. If it's lowered, it sounds like they didn't install an adjustable track bar to center the rear axle.

With a stock ride height, the rear axle should be centered. I would check that and maybe check that the rims are all the same. It could be that one has different backspacing or even a wheel spacer depending on how different they are.

Could you please take a couple of pictures and post them showing the tires and how they stick out? No rush, just trying to determine the problem.

The way the track arm works it can push the rear from side to side a bit, but you shouldn't really notice it. If you were to lift the rear off the ground you would see the left tire move out farther as the bar pulled it sideways, but it's not all that far. You might feel it move if you are hitting large bumps that cause the suspension to compress and then rebound a lot, but normal bumps shouldn't be noticed much. Looking at the pictures it doesn't look that far out but I'd still check all of the bushings.