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Is the strut bar that's over the engine held by only the 3 nuts/bolts on left and right, 6 total?
Yes
Do I take those nuts/bolts with me to the next car or do they belong to the car?
They belong to the car, they hold the strut to the car.
Do I have to consider anything while removing and replacing it, any preemptive moves I have to make?
No - simply remove 6 nuts, take bar, replace 6 nuts. Do not over tighten the nuts.
Will it screw up any alignment or balance or anything even if screw it back on with equal pressure on all 6 bolts?
No.
Are these compatible with any e46 from 2003-2005?
Yes.