"What are you going to do next Saturday?"
Translation:¿Qué vas a hacer el próximo sábado?
Days of the week, especially when it concerns something happening on a specific day or days, need the article el or los for plural. Here's good information about using the definite articles in Spanish. Scroll down to where it mentions about days of the week: https://www.thoughtco.com/how-to-use-definite-articles-3079100.
I wrote ¿Qué vas a hacer el sábado próximo? and they said I missed a word. I know I inversed el próximo with sábado but I did not miss a word. Is this something that is acceptable in Spanish?
It's OK, just not the most natural way to say it. When you place próximo after the noun it changes its meaning from 'next' to 'near', which is not that big of a difference, but we prefer to use it with years and months, not days. (e.g. El año próximo / El mes próximo).
Why is the plural form "van" not acceptable here instead of "vas"? There is no context as to whether there is one person or a group to which the question is being addressed? Still, Duo noted I used the plural form and rejected it. Any thoughts?
Two possibilities: 1) The powers to be never thought of programming it to accept a plural "you" answer, while they should. 2) Duo likes to use "you all" when they want to indicate plural "you."
Neither 1) nor 2) above is an "excuse" though. You are right that without context, they should accept both singular and plural "you" answers.