In China, take a plane means "坐飞机“, take a train means "坐火车“, take a subway means "坐地铁”, take a bus means “坐大巴车”. Hope it helps.

坐飞机 in english, you are in a airplane, actually you are sitting in a airplane. Also you can understand in this way. 坐飞机=乘坐飞机, 乘坐某种交通工具, by airplane, by bus. 乘坐飞机，乘坐公交车. Hope it helps.

Yup. A more formal way is 乘机

Because if you are not the driver of the vehicles(any kinds of vehicles, carriage, cars, planes, taxis) you are taking, you sit/坐 on the seat/座位.Though there are no seats in the ancient carriages(pulled by horses), you normally sit in the carriage not ride the horse. So we often say 坐公共汽车/bus，坐地铁/subway，坐火车/train，坐飞机/plane，坐轮船/ship，坐三轮车/(electric) pedicab，坐摩托车/motorcycle，坐马车/carriage.

你的理解稍有偏差。 “坐”用于交通工具名词前，表示“乘坐”的意思。如，坐飞机、坐火车、坐地铁等等。 而“坐”+交通工具多用于口语表达，书面语多用“乘”或者“乘坐”放在交通工具名词前。如：乘（坐）飞机、乘（坐）邮轮等等。