Thy should make the transition from marvel girl to marvel woman
It should be Marvel Woman, Phoenix or Jean Grey but never Marvel Girl. That's her teenage superhero name.
it feels weird, as Superman going by superboy. That's how it feels to me
There's nothing wrong with just Jean Grey as I feel its the name is she is most known as even above Phoenix. Jean Grey is what has been primarily used for her branding and considering they chose to name her Giant Sized issue, Jean Grey, I have a feeling its what she will officially be going by next year. I cant help but think that her use of Marvel Girl was a way for Marvel to keep the trademark active since there isnt currently an AU version of the character using it at the moment
Copyright protects someone's work. You don't need to "register" it or whatever. It's automatic: you can't copy or re-use someone's work without his permission. There is also no expiration (some work can fall into "public domain" after 70 years or more (for recent works) after the death of the author).
Trademark protects a brand. It needs to be registered and renewed. You also need to use the brand regularly to keep it registered. You can't register a name or common word because it's not distinctive enough so you can't register "Jean Grey". But you could register a "Jean Grey" logo or a distinctive character. Also a trademark must be limited in scoop (Ie it apply only to a specific domain in relation with a specific type of produces. Ie if you trademark a comic book character with the name "Jean Grey", it won't apply to a mark of pants called "Jean Grey".
Marvel didn't got a trademak on Jean Grey's logo or anything related to Jean Grey so there is nothing that can expire. Which doesn't mean that someone could make stories about a character named Jean Grey who is a redhead with super-power because that would infringe on the copyright.
You are probable confusing with the stuff about Captain Marvel: A company trademarked a comic with a character called "Captain Marvel". DC sued them because (they said) the character was too similar to Superman, which was already trademarked. As a result, the first company had to stop publishing their Captain Marvel comic.
But DC didn't trademarked their own comic called Captain Marvel so Marvel was free to create their own book and trademark it (as they didn't used a character that was too similar to one of the DC's brands).
Later DC got the right to the original Captain Marvel comic but they couldn't use the name anymore in the context of a comic book as Marvel had trademarked the character for a comic book already.