Originally Posted by Weasel

If the battery charge is low you can sometimes hear a whirring sound from under the hood from the load on the alternator, but usually don't hear the fuel pump unless the fuel level is low enough for the pump to not be submerged, which on a BMW means out of gas... so if you really do hear the fuel pump it may be on the way out.

For the coming and going balance problem, the thrust arms wouldn't cause a shaking but do exaggerate it when present. It could be the wheels out of balance, or it could in an axle shaft... though that is less likely. Best to have the wheel balance checked first as a good starting point.