Yes, depending on your choices, the protagonist and/or your partner will change their relationship roles from Top/Bottom. This will be reflected in certain lines of dialogue, although I think it's currently intended more for a future update, as it implies a sexual relationship. In any case, you can find some funny dialogue and scenes; choose what you prefer.
Regarding the rest, first of all, the introduction and notification message mention that there might be some inconsistencies because the story is being revised to make it make more sense. Now, Walter didn't forget about Gil, and yes, Gil disappeared. However, mentioning anything more would be a spoiler. I can only confirm that this is currently discoverable in the current version. Good luck! :P
Regarding the calendars, since I'm working with a home translation, I had to analyze it, and this is how it currently works. It doesn't use the traditional calendar we know, but rather its own elegant way of using the full moon, dividing it into 12 parts corresponding to each month. In essence, the dates haven't changed much, unless you're coming from an older version of the game.