I want to do a fishing chain for shiny Pokemon but I don't want my untrained Swalot in front of my party.
Yes, Sticky Hold will still work, even if the Pokemon has fainted. I had low-leveled Malamar with Suction Cups(It is the same as Sticky Hold outside of battle) go to the front of my party while it was fainted. It still worked because the ability's effect outside of battle does not rely on the Pokemon to be healthy in your party.
Hope this helped!
Yes, it does.