Does, "Sok diák a magyar iskolákban tanul." work, or is it incorrect?
It is correct but the emphasis is different.
"A magyar iskolákban sok diák tanul." - emphasis on "sok diák" - many students
"Sok diák a magyar iskolákban tanul." - emphasis on "a magyar iskolákban" - in the Hungarian schools
Why the translation is a singular student ( diák) while the english sentence is about the plural ( students) ?
Because "sok" is a number word and in Hungarian number words always take the singular.
Why are we using tanul instead of tanulnak
As of today, 23 March 2020, this is not true.