An extension can only be done when you have 30-45 days or less left on your permission to stay stamp, 45 days at some major immigration offices. So no, you'll need to wait about another couple months.
You need a very brief cover letter explaining the payment/taxation structure is as a contractor along the account statement highlighting the incoming payments. Expect to get denied unless you lead them to the watering hole.
Having a business in and of itself is not necessarily workable. You'll have to provide a scheme such as workation or remote work and also provide/create employment certification letters specifically stating remote work in Thailand is authorized. Use the group search function to get an idea of successful remote work applications.
Appears they have had success. Altho you still want to choose your consulate carefully, possibly they can supply you with a list where their students got approvals. Hopefully you can get a list of more recent approvals.
Anonymous participant 1st report on the same entry. If you re-entered Thailand then it's considered your 1st report. Previous reports don't qualify because you now have a new entry.