She's not a great actor, from GoT I'd argue that the younger cast, Kit Harington, Emilia Clarke, Sophie Turner and Maisie and the actor that plays Bran, are very weak actors. However, i do find that with weak actors if you give them a strong actor to perform against they improve greatly. That's probably what happened with Sophie/Jessica, you also see it on GoT with Kit/Peter Dinklage.
With regards to some of the plot choices, i'm reminded of a trailer review that Jeremy Jahns did where he said X-Men The Last Stand was the worst version of Dark Phoenix and, this movie decided to copy it. I think that's a big failure. I don't know if the writers were trying to tie this movie/story to the pre-First Class/DoFP timeline or just lazy. However, Magento's Mutant Camp was in The Last Stand as was Jean's childhood trauma, neither of which are from the source material.
The weird thing is they could have adapted, close to the story. Have Jean get the powers on the space mission. Have her lose control but, be good and still try to do right only to have weird Alien girl Jessica Chastain use telepathy to mess with Jean's mind and have her go Dark Phoenix. Then end it with Jean making the choice to die rather than lose control and hurt people she loves.
As for the Jean Grey School, that's what it's currently called in the comics (unless they reset it since i stopped reading). Jean and Xavier are both Dead, Logan's the headmaster and of course he'd rename the school after Jean 🙄