However, I still see the posts show on my homepage, and appear to be able to view and interact
It's quite likely that your subscribe went through, mind sharing your findings by looking at the network tab in developer console on your browser.
What does subscribe do? e.g. are my posts visible on lemmy.ml I’ve just tried posting from here and my posts are visible from there so what exactly changes once subscription completes?
It simply makes communities on other instances show up in your subscribed feed. Similar to how you join a subreddit (the difference on lemmy is that the subreddit aka community is hosted on a separate instance).
What’s the problem and how do we fix it? (Depending on the answer to #1 maybe the fix is just “ignore it”)
My suspicion is that its websocket related, I've seen similar behavior when testing locally. The best you can do is maybe browse github for potentially relevant issues and see if you can help, otherwise please just be patient while things get more stable :)