Yes it's possible, and to my knowledge, yes it's allowed as long as you're not in a popular swimming area and no where near the boat channels & pier.
I'd recommend you avoid kitesurfing there on your own and I'd only recommend it for more advanced kitesurfers. Beaches are quite narrow and lots of rocks / reef. If you get in trouble or injured there may not be anyone to help you.
go to the Eastern side of Salmon Bay on a sea breeze (SSW, SW). A fair bit of reef so you have to be extra careful. Due south wind can be very gusty here, so best on a SSW and SW.
Junction of Thomson bay and Bickley Bay on a land breeze (Easterly, ESE, SE). Usually pretty gusty and lots of reef / rocks as well. Despite multiple attempts on various days I have not yet had a single great session there, due to the wind being too gusty and the currents can be strong as well.
Be safe!
Christian - KiteBud