has nothing to do with that. Residence certificate is required to get a license regardless of your visa type. And Bangkok has issued CoR for tourists for over a year now.
You write them a sponsorship letter as the primary DTV holder and state you will be responsible for their support, and you send your banking information again.
Some embassies will require an additional 500k for each dependent, while some will be fine just your 500k for yourself.
You do not need proof of onward travel. But for your visa application, it may ask your intended dates of travel, so you'll need to give them an arrival date and a departure date within 180 days of arrival since the DTV only grants 180 days stay.
I'm going to assume you don't have any visa at all, and actually have visa exempt. You can get a single 30-day extension from your local immigration office. You would need to travel there and apply. The cost is 1900 baht, and you must attend in person (so an agent will not be of much benefit).
Umm, where is it that you saw that you need to get a blood test for a drivers license? You actually think they give a blood test to everyone getting a drivers license?
One of the requirements is a medical certificate. Costs like 100-200 baht and normally filled out by the front desk secretary at a clinic and not even a doctor or nurse.