My 2 cents.
A week is a short time to spend in Kalymnos - you will lose a day getting out there and a day back (catching the last flight from Kos may allow an extra half day) so better off trying for 10 days or a fortnight to get the best out of it IMO.
Took a 60m when we were there and didn't run out of rope on anything that I can remember but I wouldn't go any shorter than that.
Quickdraws - 12 is adequate for most stuff but one route I did required 13 so take 14 unless weight is an issue.
I was out there in September IIRC - not too bad but you want to be finding shade or getting off the crag when the sun hits it as the temperature soars.
Places to stay - Myrtees or Masouri are the obvious destinations that I know of - close enough to the crags to make walking OK - there may be others but I don't know of them.