I am a physics student just learning about special relativity for the first time, and was wondering what would happen if you emitted two photons of light from the center of a boxcar in opposite directions so that one hits and stops a clock on one wall and the other hits and stops a clock on the opposite wall. It would seem to a person in the inertial reference frame of the boxcar that the clocks should stop at the same time, as the boxcar seems stationary to such a person. However, if the boxcar is indeed moving shouldn't the clocks actually stop at different times, and why couldn't you use this experiment to determine the absolute speed of the boxcar if that is the case? I know that absolute speed can never be determined by an experiment, but I am trying to understand why it cannot be determined by this experiment.)
The apparent paradox is resolved by realizing that if the clocks are synchronized according to an observer at rest with respect to the boxcar's frame of reference, they are not synchronized according to an observer with respect to whom the boxcar is in motion. This is a manifestation of the relativity of simultaneity.
Specifically, if the boxcar is moving to the right with respect to an observer, the clock on the right wall of the boxcar will appear to be delayed compared to the other clock. Since according to this observer the clock on the right takes a longer time to receive the signal, after both clocks stop, observers in both frames will agree on the time displayed on the clocks.
See this related question and the answer for quantitative details on how much the clocks are originally unsynchronized by.
Theory of relativity tells us no. Because no matter your relative speed to other things your relative speed to light is always the same. Assuming that the clocks are in sync from the perspective of the observer in the boxcar, when it starts moving, they will always remain in sync and will stop in sync as light will always reach them at the same time. As confirmed in the Michelson–Morley experiment (https://en.m.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment?wprov=sfla1). Do have in mind that this wasnt the point of the experiment. It tried to prove the existence of aether and the earth moving through it. In failing to do so it proved that the speed of light is always the same relative to the observer. Edit: As for the answer to your edit. For the laser fired in the direction that you are moving in, it will take more time to reach the clock than the one shot in the back, however the opposite will apply for the light coming back from the clocks (light that shows you that the clocks have stopped) and because of that, to you it will seem as if the clocks have stopped simultanieously. Or that would seem like so if observed from a different reference frame. However if observed from your frame of reference, the light has no "drift" because you arent moving. That is because, for you the time slows down the faster you move if observed from the outside. However for obvious reasons it cant be measured from the inside the boxcar. To put it in other words. The faster you move in a single direction, the less speed difference there will be between you and the photon that you fired forward (observed from the outside), but because your time will slow down, that difference will be multiplied back to precisely the speed of light (observed from the inside).