I would think that since she was brought up in a royal family that she would be given the best education available, so I would believe she knew how to write, but she didn't write while on land, so in that way I am not certain.

Maybe when Ursula took her voice, she also took her ability to write because people say that writing allows to people to get their voices heard. In this way, writing could be included with taking her voice.

Not really sure, I doubt sea creatures know things such as read or write :/

What's the point of only teaching someone how to write their name?

I think it's just a plothole.

yes because she signed the scroll