good on you for financially supporting the raising of your daughter. A lot would do anything to get out of it.
I'm not going to get too personal because I don't want you, or anyone else to get the wrong idea in such a wicked world where there's so many predators, but can I ask if your daughter is a teenager, or not yet? Asking that from the angle of your daughter possibly being old enough to put some 'pressure' on her mother to allow her to visit you.
While I don't have any experience in this, nor legal knowledge, other than that, as much as you might not like the idea, politely 'threaten' the girl's mother with stopping the financial support. This support that you're providing, may have been of great help over the life of your daughter. Her mother may not want to lose it unless she has a decent income of her own. If that doesn't have any effect, you could try threatening to take legal action for part custody. How that would work between the two countries, or even if Thailand authorities would cooperate, who knows. Maybe some legal advice within your country and maybe Thailand as well, might be necessary.
I read a comment by someone recently, that said that after the initial application for a Marriage Visa, the annual extensions are a lot easier. It's only the first time that there's lots of documentation required. Is that correct?
I'm not sure about it being permanent. Besides what is stated about using a laser, it's probably not going to be permanent unless it removes the hair roots and closes the follicles forever. We all know that antibiotics will, kill and bacteria, but they can't prevent a future infection.
while the Visa Exempt entry if free & no need to apply in advance for a Visa, that Extension isn't free though, it costs THB1900. You might be slightly better off to apply for a 60 day Tourist Visa at a slightly lower cost while still in your own country if you will want to stay longer than 30 days
is correct. A Visa Exempt entry is 30 days and is free. If one opts to stay another 30 days, that Extension costs THB1900 & therefore is not free & the cost works out a little more expensive than a 60 day Tourist Visa from my understanding