Thank you a lot for your answer! The second interpretation is exactly what I expected.

@Sasha Thanks for this hint. I am not familiar at all to the theory of motives so I didn't realize they are related.

Also for question 4 I already come up with an answer by myself. Thanks!

Thanks for your reply. So in the case of weight 2, the local system L is the constant sheaf. Let j: Y \to X be the compactification of the modular curve, then the cohomology with compact support resembles the “extension by zero” of the constant sheaf on the cusps, which looks weird. Meanwhile, as shown in Donu’s answer, the parabolic H^1 resembles the cohomology of j_* L, which is again the constant sheaf in this case, so this seems more reasonable. Am I correct?

Good point, thanks, I will try to verify by myself.

Thanks a lot for your answer!