Welp, finally managed to grab both of those costumes (the ones from "other stuff" quests). I do believe you can get all of them independently of the route taken on the "main quest" and using the same character.
I got the Puppy Headband Costume at ~20 stickers and the Puppy Ear Hat at around 45.
For the grave quest, pay attention to Po's dialogue after giving enough roses (in my case: 20 beautiful flowers + 25 frozen roses + 50 witherless roses). For the first dialogue, after selecting "other stuff", he gave the Rose of Crimson Costume to me after expressing his gratitude for puting enough beautiful flowers and witherless roses in his wife's grave. Then I just talked to him again, and this time the old man thanked me for bringing enough beautiful flowers and frozen roses, giving away the Frozen Land Rose Costume.
Btw, you can use Lutie's event item machine to claim both of those costumes on all other game acc's, just like the love birds costume.