I suppose "during the last week" is grammatical but for some reason it sounds awkward. Native English speakers would be more likely to just say "last week" or "this past week". With these expressions, "during" is redundant.
Native speakers do talk this way when emphasizing a duration of time. It's fine to use the simple past with during, but again, it sounds odd when paired with "last week" or "past week".
During the last week she played the violin twice.
This would require an unusual context in which I wanted to accurately emphasize how many times she played the violin in the given period. More natural is:
She played the violin twice last week.
During feels more useful when referring to events that themselves don't describe a period of time:
During the campaign the candidate made many outrageous promises.
I heard my friend snoring during class.
It was during the war that he met his wife.
If you want to talk about actions that happen repeatedly over a period of time, simple past or present is fine. It's natural to be explicit about how often these occur.
This past week I arrived home every night at 7.
Last week I visited my mother at the hospital three times.
During this past week I kept meticulous records of every time I saw the man in the red beret walk by my house.