Kalymnos September logistics

I’m currently planning a trip for Kalymnos in early September and I have 2 big questions:
1) I found a direct flight there but I’m wondering if a small plane will be able to accommodate a gear bag + 2 carryon suitcases.
2) will early September be too hot or will the sea breeze keep things cool enough for comfortable climbing?

I haven't used the direct flight to Kalymnos, but from observing parties that have, I don't think you need to worry about whether the planes have the capacity for your gear.  Luggage fees do add up on these low cost carrier flights, though.

You will definitely have to chase the shade in early September.  Get the guidebook and study the aspect of the crags.  Plan to have as much mobility as possible (think scooter rental) so you can move from crag to crag quickly.  There is deep water soloing in Kalymnos - this would be a great option for early September.
http://www.kalymnosclimbingguide.com/deep-water-soloing-in-kalymnos/

We took the direct flight once, and there was no problem with there being enough room for checked or carry on baggage (and we had a LOT). Generally, however we fly to Kos, because we can get direct flights there from Italy, and because the Kaly flights get cancelled more often. Def. possible to climb in September - there are plenty of crags with shade for part of the day, and even a few that have shade all day.

I've taken a couple of 20KG bags on the direct flight and didn't have a problem, so you should be OK. They're a bit more lenient and nicer than budget airlines as well (basically they're not Ryanair).
September is a great month to climb, starts to cool down from summer but the water is still warm for post climb swim. You'll have all the summer crags (e.g. massouri skyline, Arhi, Ghost Kitchen) in shade until lunch or early afternoon and the north facing stuff (e.g. secret garden, Telendos north side crags) with afternoon shade. You'll also avoid the masses that arrive in October :)

We took the direct flight from Athens to Kalymnos via Sky Express. They will make you check the carry on but those twin turboprop puddle jumpers have plenty of space for all your gear.