The Church of England is the established church. As such it can marry in the religious sense and the celebrant vicar can also do all the legal stuff, hence the signing of the register in the vestry. Other churches of the Protestant tradition can do the religious bit but the marriage is not legal until the registry office one is held. I worked with someone who married first in the registry office but did not consummate the marriage until they had had the religious service in their own tradition. They were both very religious but not C of E and did not regard themselves married until the church one. I think things have changed now but not for all denominations/traditions. I seem to remember a fuss being made a few years ago by the Jewish and Islamic mainstream faiths.
As for the curtsey, Americans aren't used to it even when they come here as visiting dignitaries. They don't curtsey to the President, although I believe they stand when he is in the room out of respect for his office. Michele Obama, I am pretty sure, did not when she met the Queen but of course she was on an official visit. Perhaps Meghan thought you just did what was normal in USA.
I did find it odd that Harry, nearly 40, seemed to think his father should still be financing him.