As already stated, there are a large number of dive centers on Koh Tao, this is to meet the large number of travellers and divers arriving on the island all year round. It is possible to dive on Koh Tao 365 days of the year as conditions (although not always optimal in monsoon season) are divable and often very calm and clear. Add this to the volume of tourists and travellers and you have a year round dive industry. 'High Season' is basically June, July and August, however there is also a peak around Christmas and new year and also easter time. Therefore, I wouldn't say there's a best month to complete your DMT as each season offers different opportunities. Quieter months allow for more individual and in depth training, whereas busy months allow more team integration and real dive experiences with customers.
As for working after, this will depend on where you complete your DM, how you complete your DM, and your actions after your DM. Most dive centers like to employ from within, so check a few shops and see how it may be to actually work for them. A good DM isn't so much based on diving ability but in fact work ethic, attitude, and customer service; so make sure you are making a good impression and working hard throughout your DMT to the point where you're actually doing a staffs role already. After your DMT, make sure you forward your ambition to work towards your dive centre, and other local connections. There are a large number of freelance pros on Koh Tao, you have to look for work, you can't wait for work to find you.
From personal experience, ticking all the boxes, I would thoroughly recommend Master Divers in Mae Haad. Their pro training programmes are excellent and are majoritively taught by the course director Gaz. The staff are experienced and friendly and go above and beyond the course requirements to ensure you become the best dive master you can be and walk away as an employable and confident dive pro. Standards are met, safety is excellent, equipment is quality, community is welcoming. You won't find a better or more professional DMT experience on Koh Tao.