Desilu had a huge overhead charge that ate up a lot of budget, but was that overhead carried over when Paramount took over?
Desilu covered more of the overhead. Paramount made the show pay for all of it. So it ended up as a smaller amount of money to spend on actual production during the Paramount episodes. This is detailed somewhere in the Inside Star Trek book but I don't have it anymore to give a page number.
I doubt the writers were paid less for their work in the third season than in the previous two years; union rules tend to keep those costs pretty stable, do they not?
The cost per script is exactly the same. But there is usually a fund for 4 or 5 extra scripts that might not work out, because you still have to pay the writer even if you don't produce the episode due to the WGA agreement. Without that fund in the third season, they just had to produce any script commissioned.