MySql include partition key in index?
Multiple questions, multiple answers:


No need to "spot" potential redundant indexes: use either common_schema or Percona Toolkit to find such keys. But I've checked -- and no, there are no redundant keys.
You do not have to use sent_at in the indexes
If you do, it does not matter (partition-wise) where in the index.


But the one thing that is not told is how you query your table. Do you do an equality check on most fields and range-check on sent_at? In such case better have sent_at indexed last. If there is an equality check on sent_in or range check on same field, with little or no filtering on other columns, then have sent_in indexed by itself, or first in sequence. Right now you have 1 month of data in each partition; getting it indexed by time is not such a bad idea.