下用在动词后面：1、表示完成或结果，留下，放在那里了，记下了，写下了。2、表示继续或趋向，走下去（继续走），停下（趋向停了）

I suggest you learn Chinese as a baby, not student, cause it gonna be very hard for a foreign speaker to understand Chinese grammar. like 下, it can has many meanings, e.g. lower, next, enter, action of stop, conquer, remove something from somebody, put yourself into, draw a conclusion, measure words = times, less than. I didn't write all meanings of 下. To be honest i dont really know how to answer your question, because my Chinese teacher didnt teach me when i was a child. She did teach me 留下 记下 as phrases and they are high frequently phrases, just remember it. as 停下50辆车, you can also say 停50辆车 which we we use a lot more in spoken Chinese. sorry for poor grammar :)

it's an intention, towards sth. same as 'write it [down]',

They are abstract. I think they are similar to English as in "write down; catch up with;". It can be sensed, but not explained