A small bit of side-to-side play is actually pretty normal for the rear axle shafts. The retainer clips inside the differential have a little bit of space to wobble, by design, and so the only thing that would keep the axle shafts from sliding back and forth a bit because of this would be some sort of rubber washer/cushion. Some Explorers actually had these washers from the factory, and some aftermarket axle shafts had them, or you can buy the washers/cushions, but it's a PITA to do unless you have to because you are replacing the axle shafts or changing gears and taking apart the rear diff anyway. Plenty of Explorers don't have the cushions and the axle shaft play isn't an issue.
The vibration at highway speeds is often something like poor wheel/tire balance, or tires that aren't perfectly round. Unless you're doing the balancing yourself, you're trusting the installer to care about getting a perfect 0.00 reading on the machine, and to double-check their work by spinning it one more time. Usually this doesn't happen and you just get wheel weights pounded on "close enough" to the position the machine indicates.
Also, as tires wear, the wheel/tire balance is affected. Not having the them re-balanced every 5,000-10,000 miles or so can cause just the right amount of off-balance vibrations to be annoying at highway speeds. This is a good reason to get tires from a place that offers lifetime rotation and balance for as long as you own the tires.
Other vibration causes include rusty, flaky rear driveshafts that are off-balance, or a driveshaft that has lost it's balance weights, poorly aligned front wheels, and either hard, crusty stock rubber bushings or bushings that are shrunk or even falling apart and allowing play instead of being a solid pivot or attachment point.
