Yosef Karo

“ And it is permitted to play with a ball in the public domain even though this is just recreation (Tosafot and Rabbenu Yerucham) . If there is an eruv, it is permitted to carry and move everything that is a useful object, even something that is not at all related to a holiday need (Ra"n chapter 2 of Beitzah) . ”

“ Paragraph 8 - We take wood leaning on the walls of the Sukkah and burn them. Rem"a: This is also the case if one threw bundles on top of the schach because they are not legally absorbed; if one had the intention to burn them, he is permitted to take them from there. ”

“ Paragraph 2 - It is forbidden to move anything [between domains] on a holiday for the sake of a non-Jew. Rem"a: But it is permitted to send something to [the non-Jew] by means of a non-Jew, like if one has to return a deposit to an angry non-Jew or similar situations (Mordechai the end of chapter 1 of Beitzah and Beit Yosef in the name of the Tosafot chapter 1 of Ketubot) .
סעיף ב - אסור לישראל להוציא שום דבר ביום טוב לצרך אינו יהודי. הגה: אבל מתר לשלח לו על ידי אינו יהודי, כגון שצריך להחזיר משכון לאינו יהודי אלם וכדומה לזה (מרדכי סוף פרק קמא דביצה ובית יוסף בשם תוספות פרק קמא דכתובות) . ”