In English "before" can be used to mean "in front of", but it sounds odd in casual contexts. In my ears (as a native speaker) it implies that two things are in each other's presence and are 'facing' each other, or somehow that there is an important tension between them. Like, "I walked in and she was standing before the king" immediately draws up the image that she and the king are facing each other and interacting somehow. Whereas "I walked in and she was standing in front of the king" could mean that, or it could mean she's facing the same direction as him, looking at me, but is standing closer to me than him.
Other examples: "The Empire State Building loomed before me"--there's the tension of me looking up at it and experiencing it. "Before you is the newest invention at Renham Industries." -- in this one the point is that everyone is looking at it and it is the center of their attention.
And of course, "before [someone's] eyes" is a set phrase.
So in this case, "The car stands before the bridge" sounds a little bookish or archaic, since the car and bridge are not interacting at all. It's not wrong, per se, just odd. (Unless, of course, this is another sequel of Beauty and the Beast where the Bridge and Car are both people, and the Car is asking the Bridge for help for something.)
It's not necessary, but using it here is perfectly normal. You can omit it if you want. However it also serves to point out that the car is in fact not moving, because if you say "перед мостом машина" it could be interpreted as it was driving by
As for the "car wouldn't be "standing" it would be "parked" part, there is a word "припаркована" but it's a bit too specific term, implying an intent on the part of a driver. So if I park a car near the bridge and later I need to explain where it is, I'd say "моя машина припаркована у моста", but a casual observer is more likely to simply say that the car is "стоит". They can use "припаркована" as well, but they probably won't, unless they want to comment on the driver's choice of a parking place. There's a fine line here, I'm not sure I can adequately explain the difference.