Is carrying permitted for no purpose at all or it needs to have a holiday purpose. If it's permitted for no purpose at all why is it forbidden to carry for the non Jew

Carrying And doing other works on the holiday not directly for food but for other holiday needs.

Sending a non jew to make a purchase on yom tov. Telling a store owner to fill a utensil. Not measuring food for animals

I said I that's mostly non Jews we assume it was brought for the non Jew. A question if it came from outside of that their home if they going lives in the town and these things are common in the town it's permissible. Something that came from outside of the techum on the 1st day of Rosh Hashana is permissible on the 2nd day

Something that couldn't be picked or trapped and came from inside the תחום or from outside of תחטם or can use it and who can move it

A non jew bringing a present to a Jewish person from things that might have been trapped or picked on 1st day of Rosh hashana, some say it's permissible and some say its forbidding

If the non Jew picked or trapped for himself it's still forbidden on the holiday but it's not required to wait after the holiday enough time to do the action. In a case of a doubt whether it was picked or trapped on the holiday it's still forbidden it on the holiday.

Something picked or trapped on 1st day of Rosh Hashana is asur until after second day , plus enough time to do the action. Same thing when Yom Tov is Friday or Sunday.

Something that was picked or trapped on yom tov is asur until after holiday and enough time to pick or trap it. If picked on first day of holiday, it becomes mutar on second day after בכדי שיעשו.

Opinion of Rema that one should not use thing that were caught by a goy on first day yom tov , until בכדי שיעשו after second day. Unless for guests.