"I eat sushi with chopsticks."
it's helpful to think of で in this context as meaning "by means of." so, "I eat sushi by means of chopsticks." it indicates by what manner you're carrying out an action.
(though it can also in other contexts indicate in what place you're carrying out an action -- 公園で歩きました means "I walked in the park," as in "I walked, and the park was where it happened," while 公園に歩きました means "I walked to" the park, i.e., "I walked, and the park was my destination.")
お箸・箸 is the word you'll always use/hear to refer to chopsticks.
おてもと is an abbreviation「お手元箸・おてもとはし」literally meaning "On hand chopsticks". These are chopsticks for personal use to eat with (as opposed to お取り箸 that are chopsticks used to serve food from one plate to another). It is a bit old fashioned/archaic and almost exclusively used on the paper wrappers of disposable chopsticks now.