Here are some answers to your questions:-
Will you get ill? Possibly, but then you can get ill at home. You can minimise the risks, by drinking bottled water, avoiding ice cream, ice cubes and salad. Eating hot foods which you have seen to be cooked. Can avoid market stalls, but that is part of the experience!!
In terms of malaria, I wouldn't take anyone's advice from here. As everyone will say that they have been to Thailand, not taken malaria pills and not got ill. I suspect that they won't be around should you get recurrent malaria. Check out this website for information on tropical diseases, it is run by NHS Scotland:-
http://www.fitfortravel.scot.nhs.uk/
You will see that the malaria map indicates a pretty high risk of malaria over the country. If you still have concerns, I would go and see a doctor who specialises in tropical diseases. Your local GP should know of one.
From memory the very high risk areas are the north and border areas with cambodia. I have been to most of Thailand and not taken malaria pills, but I would do your own research on this area.
There is lots of food for vegetarians. All of the rice and noodle dishes can be done without meat. Tofu (yuck) is used extensively in cooking in thailand.
Can you fly direct to Phuket. It doesn't look like it now. Phuket Air used to fly from London to Phuket, but then they got black balled by the EU for having a poor safety record. You may be able to get a charter flight. You will have to fly from London to Bangkok, airlines which fly this route include British Airways, Thai and Eva. All of which are pretty good airlines.
Once in Bangkok you can either fly or get the bus. If you fly your options include Thai and Air Asia. The flight is about an hour and should cost about �30. If you get the bus it is about 12hrs by night bus