, Thailand does not have great success with its online services. The eVisa website was developed by one of the big banks. But of course, it won't be a big challenge to develop a such system for the TM6 for a professional software company.
, do you mean if the minors are not your own? In general, you might need some kind of Power of Attorney from the parents if you are traveling with minors not being your own. It is most likely at the airport in India, they will ask for it. If you as a foreigner are leaving Thailand with a child with Thai citizenship, they will most likely ask for it.
Try contacting the Thai Embassy in India. Unfortunately, the website does not seem to work properly. Usually, the rules for minors are the same as for adults as long as we talk visas on arrival and tourist visas. When talking about long-term visas, the rules for dependants are quite relaxed.
Most immigration will like you to make your application about 1 month before expiry, but go early, get the application form and updated list of requirements, and ask when to apply.
On a Non-O visa, you just need a place to stay and some money (20k THB). Actually unlikely that they will ask to see the money. If it is a single entry Non-O they might want to see a return ticket (or onward ticket). All Covid restrictions have been liftet.