The tires don't matter for the alignment and you always balance tires off the car.
When aligining a front end the alignment system is takes it's measurments off of the wheel spindles (or hubs) so the wheels and tires are pretty much irrelevant.
There is also no way to balance wheels/tires once they are on the car. You'd be balancing the car - not the wtires. That big garabage bin you saw was the balancing machine.
The big plastic shield (usually black or grey) is a safety shield so that anything flying off of the tires doesn't hurt the mechanics. (On occassion, if someone doesn't know what they are doing, the entire wheel and tire comes flying off during balancing! Doh!)
And yes, you should retorque your lug nuts after driving for a bit but you know what? Nobody does it. Very, very few garages even measure the torque when they put the lug nuts in to begin with and most people don't even own a torque wrench (and even if you owned one, you wouldn't spend $300/year to keep it calibrated properly). Just check you lugnuts and make sure they are all snug.