One gotcha to be careful with is that if you run a query spanning multiple partitions, it will run them all at once and if your database isn't super big - will bring it to its knees.
Outside of that really no issues. We also use Timescale quite heavily, which also works fantastic.
Not if these are analytical queries...
If you're wondering what the point of partitioning a table that's going to be used for analytical queries: if you partition by insertion time, and your table is append-only, then you're only ever writing to one partition (the newest one), with all the partitions other than the newest one being "closed" — VACUUM FREEZEd and then never written to again. So, rather than one huge set of indices that get gradually-more expensive to update as you insert more records, those write costs will only rise up to the end of the month/week/day, and then jump back down to zero as you begin writing to a new empty partition.