Hi and welcome to Holiday Truths.
Most of your questions are weather dependant. The last two weeks in September can be very warm, even hot during the day. Can get a bit chilly on a night.
If the daytime weather if not too good, the more things are closed, but -
Will we still enjoy ourselves?
Rather down to you, there are places to go to like Nessebur, Bourgas, and the various excursions on offer, or even get to know a friendly taxi driver who for a modest price will take you to the real Bulgaria away from the coastal 'Disneyland'.
Peter