This is the English subjunctive. It takes the infinitive form without "to" It is important that the students be silent during the movie. There are other forms. Take a look at https://www.lexico.com/grammar/when-to-use-the-subjunctive

Yes, I think that's right, but also know that English-speakers might not know the right verb either... If you were to say GOES in conversation with someone, I don't think it would sound wrong to a lot of people. And I can definitely imagine native speakers saying GOES, too.

I think in American English this would be split into two sentences, one a statement, the other more traditional, like so: "It's important. He needs to go to the hospital." Or you could vice versa those same two sentences so you get the exclamation point at the finish.

Hi- it’s correct. It’s one of the rare cases where English uses a subjunctive form. However, many native speakers would ignore this, especially informally, and use “he goes”. There are a number of similar verbs followed by ‘that’ clauses where subjunctives are used (e.g. suggested that, required that). Hope this helps!

Now I'm confused. Is GO correct or should it be GOES?