1. What did you do during the summer vacation?

2. What did you do during summer vacation?

(Which one is correct, #1 or #2?)

Both are correct, but # 2 is used more often.

Thank you?

What about the following?

1. on (the) summer vacation.
2. over (the) summer vacation

(Are both all correct?)

Grammatically, all are correct, but with the word "the" included, the sentences would read awkwardly. Better to omit "the."

Both sentences are correct; however, there is a slight difference in the placement of the word "during." In sentence #1, "during" is placed between "you" and "the summer vacation," whereas in sentence #2, "during" is placed between "summer" and "vacation." The difference in word placement does not significantly change the overall meaning of the sentences. It is a matter of personal preference and style.