To breath they open and close their mouths to force water over their gills.
And, as you can see, they can be fairly sociable. They can be fairly territorial too, but it’s easy for them to get used to other eels, or even entirely different animals; they can be tamed and love to be petted. Some divers “befriend” wild ones who come to greet them and get cuddles even for years.
The bigger ones can easily bite your fingers off, sure, but the same goes for some dogs. They just don’t know their strength :(