箸 and 食器: a specific utensil and tableware
Distinguish counting and placing chopsticks from the collective term for tableware.
Counting chopsticks
箸, read はし, means chopsticks. 箸を一膳用意しました。 means preparing one pair of chopsticks.
箸が一本落ちました。 says that one chopstick fell. Distinguish a pair counted with 一膳 from a single stick counted with 一本.
Referring to tableware together
食器, read しょっき, collectively refers to vessels or utensils used for eating. 食器を棚にしまいました。 means putting the tableware away on a shelf.
食器を洗いました alone does not identify the types or number of items washed. A specific list requires surrounding information.
A tool and its location
箸で豆をつまみました。 says that someone picked up a bean or beans with chopsticks. 箸で marks the tool, and 豆を marks the object.
箸を皿の横に置きました。 means putting chopsticks beside a plate. Do not change 横 into inside or on top of the plate.
Does 箸が一本落ちました mean that an entire pair fell?
Show the answer and why
No. It says that one chopstick fell. 一膳 counts a pair.