Here's an ex-technicians .02:
You NEVER mount a tire with the balance dots out of alignment with the valve stem. As a shop A-Tech, I had other techs re-mount tires when they did not pay attention to the balance dots.
Manufacturers mark the spots where there are lighter segments in the tire. This removes the chance of having excessive counterweighting/bobweighting. It also allows a "seasoned" tech to rapidly balance a wheel. You can safely bet that 180 degrees from the valve/dot will be the most likely spot for a few grams.
It is acceptable to mount a tire with the balance dot out of position IF the tech has used an inordiante amount of weight to bring the wheel into balance, or there is a need for over an ounce in one particular spot. If it is necessary to re-align the balance dot, the tech, service advisor, or CSR should note it on the receipt, and a new mark should be indicated by using a tire crayon.
Most of the time, if a tire on a known good wheel needed more than 2 ounces, we would return the tire to the manufacturer. Safety and quality first, and foremost.
I have seen where improperly mounted tires needed an amazing amount of weight to bring them into balance. After remounting, they needed a nominal amount of weight. I have also found that the better the tire, especially the premium tires, the less balance work needed. The worst I have seen was a set of Kenda tires. I mounted a set by hand. That's right, NO TIRE SPOONS. That's a scary thought. How secure could those beads have been?
Static balancing has always worked well, and always will. BUT...Dynamic balancing, when done right kicks butt. The only down side is that dynamic balancing more or less "tunes" your wheel to the R.P.Ms of the balancing machine.
One final note, use the SHORTEST valve stem that you can find, and easily use. Longer stems will actually stretch (yes, it's true) just a bit at warp speed. That won't cause you to lose air, but for you anal-retentive types, it will affect your dynamic balance. Don't worry, it won't hurt your handling, it's just a piece of tech info. that I wanted to pass along. Stay away from valve angle adaptors, valve extensions, and PLEASE, don't put those damned "Tire Flys" on your valves.
Oh, and PLEASE tell the shop if you used Slime. Also let them know if for some reason you felt compelled to use Fix-A-Flat. That stuff is dangerous if it is vented into an environment where sparks, or other sources of combustion are common.
SO...If it were my tire, I'd smack the kid that installed it across the knuckles with a ruler, and show him or her the light.