If it is a rear wheel (bent, or out of balance) you will usually feel it more in the seat than the steering wheel.

Yes, weights can fall off. They may have not noticed a bent rim if they weren't looking for a problem.

If the tires aren't round, it doesn't matter how well they balance them, they will still shake. That can happen, particularly with inexpensive tires.

The tire store you dealt with should be able to balance them. If they can't, I would go to another tire store and pay to have them balanced. Look for a shop that has a balance machine called a Road Force balancer, it is more sophisticated than a simple balancer.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue