"He will not play tomorrow."
Translation:Li ne ludos morgaŭ.
Is the position of 'morgaŭ' flexible? I put 'Li morgaŭ ne ludos' but it wasn't accepted.
Yes, that should be correct. Many word order variations are still not accepted for the course, which makes sense given how many possibilities there are in Esperanto. Feel free to report it.
Is "li ludos ne morgaŭ" truly incorrect, or just not accepted here? Is there actually a rule dictating that the negation must come before the verb?
Yes, there is a rule that "ne" stands before a word which it negates. So you negated "morgaŭ". He will play not tomorrow.