I have an input or suggestion for the OP. I really don't care if I get any thanks for it. OP can thank other forum members who might comment about it.
Here it is: You should support not just your daughter, but also the mom. Why?
Many Filipinos try to go abroad to find work. Your daughter’s mom might also be thinking about it. But if you give support to the mom, she might consider to just remain in the Philippines, be a stay-at-home mom and take care of your daughter. The cost of supporting her might even be cheaper than the cost of childcare in the US. (How much is daycare now per hour?)
When both OFW parents go abroad, they leave their children back home under the care of relatives such as grandparents, aunts, uncles. Sometimes, these relatives don't care, even though they're receiving money / remittance.
As another forum member had put it on another thread, the children are put on "auto-pilot". Children grow up without authority figures. They get mixed into bad company, become lazy, have no motivation, get into trouble. When the parents come back, they find it a nightmare to undo, and it might not even be possible to undo, several years of bad habits.
So, I think it would be best if your daughter’s mom would be the one to take care and supervise your daughter, with you “co-piloting.” Any parent would want to see their child grow up with them. But sometimes, it just can’t be. And one just has to accept it, and work out a plan that will work for all.
I have an acquaintance, Filipino married to a foreigner, who is a stay-at-home mom. She and her husband met and worked on a cruise ship. They were based in UK. They can’t both work because there will be no one who could take care of their very young child. And with the cost of living in the UK, they can’t afford living there with just one income.
The Filipino mom and the child now live in the Philippines, where the cost of living is less. The dad visits around every 4-5 months. But in between, they do a lot of face time over the internet. The mom is focused on her child’s upbringing, education, and teaches her good manners and respect especially to the dad. This works out well for them. The mom said they’re planning to go back to UK in the future.
Once your daughter has grown up, she might want move to the US for better opportunities, perhaps for a better life. And she would probably look forward to that as she matures, and will be thankful that you gave her a good childhood, an education and the option to live in the US. When she’s there, it’s your turn to be with her (from around age 22 onwards) to guide and teach her about arts, culture, life.
My take on your 4 questions:
1. If your daughter is a US citizen, she can move to the US either without any or with very few restrictions once she’s an adult. But if you want to bring her to the US as a minor, you would need consent from the mom for her to go to the US.
2. Similar to question 1. As an adult, she can. As a minor, no. You need the mom’s consent.
3. Remember that your daughter is also a Philippine citizen. The mom can apply for a Philippine passport for her daughter. Unlike for US passport where 2 parents are required to make an appearance, for Philippine passport, I think only 1 parent is required. I’m not sure as to whether the mom can bring her daughter to Taiwan with her without your consent. But I think it would be easier for her to get that passport if your name is not in your daughter’s PSO birth certificate.
4. Sometimes, the embassy intervenes when it’s a dangerous or life-threatening situation. But in your case, the embassy will probably tell you that they don’t offer legal advice, that you would have to talk to an attorney.
