The only visa that will allow you to do work in Thailand is a Non-B visa, this requires a plethora of paperwork from the Thai company, and even if you have this visa, you are not allowed to do any work until your work permit has been approved, more paperwork, and this can only be applied for when you are in Thailand
You cannot apply for an ED visa just out of your own, your first step is to find a school, they will provide you with the appropriate paperwork, only then can you apply
It will only be extended until your work permit expires, but you can apply for the extension anytime you have less than 30 days on your 3month visa left