Letter completion: In my time doing this exchange (I think this is my third time), I've had my letter finished before sign-ups closed twice, and the third time tried to get it done ASAP. I agree that Proposal #1 would be burdensome for participants; for some like me the sign-up process can already burn through quite a few spoons so I might either forget to do it or it might discourage me from finishing because the ADHD brain knows it's another thing to do.
Proposal #2 interests me. If I were a participant in a situation where my recipient provided an unfinished letter, I would greatly appreciate the chance to switch to someone who has a complete letter/more interesting prompts I'm confident I can write!

Check-in period: I have never missed one of these, which in some ways surprises me but tbh I think the discord server has helped a lot in that regard since afaik there's always been a reminder to check in. I like the idea of making the check-in optional for the reasons you mentioned. I know personally it's always been exciting to get to share a sneak peek of what I'm working on (esp since lately it's included some sort of art), so I think it would miss it if it went away entirely. I especially like your proposal since it avoids the unnecessary PH as you mentioned!