Hi Nick and welcome to the forum. As for your question on when to arrive, normally December / January are a good time to arrive, then you can travel a bit in Thailand, then either start teaching summer school in March, or job prospect for the new school year in May.
BUT, most of Bangkok is now either flooded, about to flood, evacuated or inaccessible. It will take months, if not longer to clean things up. So far, Suvarnabhumi Airport is still operational, but, once you're there, it would be difficult to access other parts of Bangkok.
Chack out any updates on Bangkok flood latest news, Thai flood, Flood Map, Thailand news, Breaking news, Stock market - The Nation
As far as contacting schools, it's far better to arrive first, take a look around, THEN either call them or go in person with your CV / resume. Thais avoid e-mail like the plague but they do respond well to face to face meetings. If you apply at a language school, the Director of Studies would be western and a native speaker.
As for the insulin, you could contact any of the hospitals. There are many private hospitals in Thailand and they're good. Most services and prescriptions cost far less than in the west, or in North America at least. since you'll be brining a year's supply with you, you'll be OK contacting any hospital that you come across or live near once you have aplace.
As for teh visa stuff, I left Thailand almost four years ago, so unfortunately I can't help you there, but I'm sure someone who can will come along soon.