Any reason we can't say "The spring is very beautiful"?
That should be accepted as well - it's perfectly fine.
Is there some reason why "Spring is very pretty" is unacceptable?
I think pretty is the wrong word here, should have been nice or beautiful.
"Pretty" or "beautiful" are commonly used. "Nice" does not convey the same emphasis.
Why insist on "the" before spring?
Does this mean spring in general, or this particular spring?
It means spring in general.
Question about word order.
Would this be okay? A tavasz(topic) nagyon szép(focus)