Thai immigration do not regard a one day overstay as an actual overstay as they are aware that minor problems can occur to travellers causing small delays. I have a few one day overstay stamps in my passports & I was never charged for the extra day.
I did my son’s passport and had it back in 11 days but they have to tell you 6 - 8 weeks to allow for peak times, reduced staff due to holidays, public & bank holidays etc.
A motor vehicle is necessary to drive in Thailand but seriously an international driving licence & your normal driving licence from home will suffice with local police both in cities & in the rural areas. I’ve shown just my home driving licence several times & it was accepted.