This is correct. You only need birth certificate. Birth certificate itself is the proof of who the legal father of the child is. What matters here is the "legal" father, not the "biological" father. Court document, DNA test, those are not required.
You can get a 90 days Non-O visa based on relative (i.e, your child) in your country. Visit Thailand on that visa, open a bank account, put 400K Baht in the account to season it. Then marry your girlfriend, and apply for 1 year extension based on marriage from within Thailand itself.
Don't expect them to fix it. They'll feign ignorance and say that it must have been you who entered the wrong details. They won't even go back and check in their system. They'll just tell you to reapply. But I wish you good luck. Let us know what happened.
Also you would only need to go through the hassle of reporting TM30 and 90 days if you plan on extending another 180 days in thailand. If you do not plan to extend, TM30 and 90 days reporting is not needed. You just leave the country before 180 days, nobody at airport immigration checks if you have filed TM30 and 90 days report or not during your departure from the country.
If landlord or hotel owner fails to file TM30 in time, it’s the tenant that gets fined when they visit immigration for extension/renewal. It’s so ridiculous. Why the average tourist and tenant have to pay the fine if it’s the landlord/owners responsibility to file the TM30? No action is ever taken against the owners, so why would they go out of their way to file TM30 for you? Hence you hear stories such as this all the time.
Most probably if overstay is only for couple days, you’ll only need to pay fine and they’ll put a overstay stamp in your passport and in their system. You’ll not get banned, but acquiring some future Thailand visas might become more difficult and they might ask you to come for an visa interview to explain your case before granting/denying you the visa you applied for.
Or you can try the Thai Soft Power route (Muay Thai, Medical Treatment or Cooking Classes). This would get you the DTV, and would cost much less than any agent's fees. Agents are quoting upwards of 100K THB for DTV now a days.
You're applying in Malaysia? I would highly recommend you stay away from that Embassy unless you have solid paperwork. It's one of the most difficult Embassies to apply from, that require a lot of documents and all documents needs to be stamped and verified from Ministry of External Affairs or your home country's Embassy. You'll have better luck if you apply from HCMC, Vietnam or Jakarta, Indonesia or Phnom Penh, Cambodia. If you want faster processing time, visit Taiwan but be ready for your documents to be scrutinized carefully and be asked a lot of questions in Taiwan.
You're right, but that's how Thai embassies work. You won't be able to have a logical argument with them on why freelancers don't require "remote working" permission when they ask you for this document. Although, I've heard people getting away with only providing Employment/Freelancer contracts as long as it mentions that you've permission to work remotely. Depends on the Embassy you apply to. But it's better to have this document rather than not have it.