I think the 12-35mm is well worth its money compared to the kit lens.
For primes I would consider the Panasonic Summilux primes or Rokinons.

Usually folks who claim the gear does not really matter much have a fortune of gear to their disposal. I think rock bottom gear hinders the process, middle of the road gear no longer does, high end gear is technically and reliability wise the best but it slows you down enormously.

Next step should be to add lighting to the indoor shots, you need light not only to overcome noise but to add "depth" to your shots.

I believe you would be better off to start with a varifocal lens, then add other cameras and primes later.

You write:

"I was an unpaid producer, DP, and editor for a short film. The writer was also the director....."

Hard to tell with this little information but wat you write seems to imply that someone asked you to do something unpaid, the one who asked would technically be the producer, so if it was the writer who asked then he is factually the producer of the video. If there was an understanding you worked for him or his production, even when not payed, the rights belong to the producer.

Looks technically superb!

However when I view the video I find the technique too much in my face. I think it should always be the content that should come first. Don't get me wrong, content is definitely there but I feel it has a second place.

"More matter with less art" as they say in Hamlet.

I hope you find the critique constructive.