After playing MySekai for 3 whole days, I will share the info I've got so far from directly playing: (Only a little long, I hope it helps people who've just started out! 👍)
I don't think you can share/sketch music records unfortunately.. (darn) and you get music only when you unlock the final area at Rank 5 (double darn).
Supplies replenish twice a day at a certain time (you can view it view via the forecast thingy you can click on when you hop in-between places.)
You can get different sets of dialogues from characters depending on the time of day (once in the morning, another in the afternoon, and one in the evening.)
Characters from a certain group's door will change at least 2-3 times a day (example: Rin and Airi could be there in the morning, but Shizuku and Minori could join in later within the same day, usually by switching out with one of the existing visitors.)
If you change the door, I'd recommend doing it before any periods of daily dialogue changes so you can get Memoria from them right on the day you did it. (I changed my door before evening and strangely got Memoria from the new characters at like midnight. Idk how consistent that is.)
Here's the order of importance I've gathered from what stuff you'd want to pick up the most: Items that appear only under specific weather conditions, red tool boxes, cotton/linen plants, ores, rock, and lastly wood since you have so much of it available. Idk what the last area offers yet, so this may change.