I have started a discussion on GitHub about adding a confirmation when a student clicks "Grade test" (https://github.com/openwebwork/webwork2/discussions/1612).
In terms of saving answers, I think it is good to tell students that their answers are saved when they switch pages on a gateway quiz, though I don't know if this should be included directly as part of the Gateway Quiz page.
For my usage I have switched all of my quizzes to one question per page so that answers are saved every time a student navigates to a new page. I have run into the situation you describe where students submit their quiz after answering the first question, but I have mostly mitigated this by providing them with a practice quiz in advance of the first quiz, and then having a low stakes first quiz where they are not significantly penalized for submitting after the first question.
I also am going to suggest that in the Gateway pages, the links for "Preview answers" be changed to "Save and preview answers", since clicking preview is another way to make sure that answers are saved.