Walk into any K BANK or Muang Thai Insurance anywhere and talk to their officers there. Ask them for the foreigners accident plan under Muang Thai Insurance.
afaik, no. It's better for you to get a separate one. My reason is simple, if anything happens to you (touch wood), it will be easier for you to go to any hospitals and seek treatment etc. I mean if you can afford it, medical is expensive everywhere so having an extra one doesn't harm us.
Now if you're looking for an additional coverage for yourself, I suggest you to get a separate one for foreigners and pay yearly. This will come with English terms and conditions and you will understand better for what you're covered for.
the thing is about this "health insurance" they are compulsory for anyone who's employed in any Thai companies here. The coverage is not extensive as you may have hope for as you got to choose your main hospital which you opted for and not all major hospital qualify. Mostly governmental hospitals etc.