There's no requirement to have ANY amount of time on your passport to apply for a 1-year extension (not a renewal). If you don't have enough time on your passport for your extension, you'll just receive a short extension that's only valid until the expiration date of your passport.
There is no hospital that is or is not authorized to provide medical treatment and issue a certificate. There is no certificate.
You need them to provide you with a treatment plan, and you submit that for DTV. It's up to the embassy if they accept it or not. There's no authorization for any hospital to qualify for DTV.
Actually, every country that qualifies for the 30-day bilateral agreement visa waiver, is also on the 60-day visa exempt list. It's weird, but when they released the new list of countries for the 60-day visa exempt, the bilateral agreements stayed in place. So there's no longer any country that must be limited to 30 days. There are still some countries only eligible for 15 day Visa on Arrival though.
The first photo is partially true. The requirement for proof of funds if entering as a tourist has been a rule for decades. It's very rare to be asked, but if they ask and you cannot show the cash, they can deny you entry.
There is also proof of onward travel required, but not within 30 days. It needs to be within the number of days you'll receive when you arrive, so likely 60 days for a tourist.
The 2nd photo was created by a visa agent, and was only relevant for Koh Samui.
There are very few hard and fast "rules." Entry is at the discretion of the immigration officer you are standing in front of, so whatever they decide are the rules for that entry.
Yes. You can only activate a visa issued by the embassy when entering Thailand. So if you received it after arrival you would need to leave and return.
If that's what you are interested in doing, the best thing would be to make a brand new post stating where you are in Thailand and what you are looking for an agent to do, for example "Looking for an agent to convert to non-O visa from tourist visa in Bangkok, including opening a bank account"
Something like that
Though the place with the most agents, and will therefore be the cheapest because of competition, would be Pattaya.
One of the requirements to get a non-O visa inside of Thailand (if your immigration office offers these) is that you must already have the money in a Thai bank account to show. If you don't have a Thai bank account, you cannot get one as a tourist. So your option is to leave Thailand and apply for a non-O visa from a Thai embassy, or to pay a lot of money to an agent to who will help you open a bank account and convert your tourist entry to a non-O visa. This will probably cost like 50,000 baht.