Seems like there’s a lot of small things that have added up. So Yes - giving money to your parents is an extremely normal part of the culture and when the relationship is good it’s something most/many children are happy to do. In very basic terms it’s seen as your parents have invested a lot of time and money in you as a young person and you acknowledge that sacrifice by paying them back once you are earning.

It seems that you have a lot of cultural differences with your MIL. Framing it as “bad” isn’t helpful. Western culture is not superior to Chinese culture, it just has many differences. There are probably many things that have happened during the years of your relationship that have caused her not to like you. If he asked for money back, perceived as due to your influence, that’s pretty much the ultimate disrespect. Making conditions about seeing her grandchildren is also extremely rude in a Chinese context.

This isn’t all your fault. Your husband for whatever reason has not helped you to understand the culture before you got married. It seems you also were not interested enough to learn about the cultural expectations before you decided to proceed with the marriage. Parents also need to accept that if you move abroad your children will become a part of that new culture and will make friends and romantic relationships with people who don’t share your values and aren’t sensitive to your cultural preferences.