It can’t be done within Thailand but you can use a nearby embassy/consulate. You have to be in the country you are applying at for the application and until the visa is issued.
You can apply from a nearby country. You don’t need to return to France but you do need to be in the country you’re applying from even though it’s all online applications these days.
There is no limit on the amount of time you can stay out of Thailand. You can come and go as you please within those 5 years. The only limit is 180 days per entry. Possible to extend but it looks way easier to bounce as needed.
That can be the case, but a reputable one will want you as a repeat customer or refer others. That won’t happen if you aren’t happy with their service.
You ask for no agents but it’s those that can absolutely advise you on the best policy options. What works for Joe Bloggs and his personal circumstances may not be the best option for you. That’s why health insurance agents exist. To advise on your best options. They don’t charge you for their service. They get a commission on the policy you decide on.