I usually opt to keep it with me, and stow it in the overhead compartment. That way, I don’t have to leave space for it in any of my bags; it’s just considered part of my clothing that I’m wearing when I board. If it’s extra cold for some reason, I’ll keep it at my seat.

Overhead compartment or under my seat.

I roll it up into a bundle and use it for a pillow, or lay it across my lap as a blanket.

I would carry it with me and use it as a pillow or stow it under the seat. Although if I was travelling to somewhere warm I probably wouldn’t take it at all (and just suffer the cold on my way to the airport).

I wear it onto the plane and store it in the overhead compartment or continue to wear it if the plane is cold. Sometimes I take my arms out of the sleeves and have it just over my shoulders and sit back on it. I try to wear a coat that is not too bulky, but I rarely visit places that are well below zero centigrade in the winter. I usually can get by with a typical wool coat. I make sure to pack clothing I can layer.

first choice would be the overhead compartment unless of course other have grabbed it first to stuff more than allowed. (I hate that!!!)
OR if you have room you could stuff it in your checked luggage or carry on. Whey not just dress in layers and not lug around a huge parka?