It is a very good question. And also a relatively complicated question. About your second sentence，maybe we can do a little correction. 我的手机在柜子上，请帮我把它拿下来。 We can use 它 to substitute 手机，and the verb we had better use 拿，because 手机 is not that heavy，no need to 搬，haha. This sentence is surely right，and 下来 here uesd after a verb means from the upper to lower. The first sentence，学生总是安静不下来。安静 is used as a verb here，下来 indicates one action is finished，loses its original meaning. so 安静下来 means stop the state of noisy and keep silent. If we add a 不between them，means the students can not stop making noise and keep silent all the time. We have many similar expression like this, for example, 车停不下来. The car can not stop. 衣服太紧了，脱不下来. The clothes is too tight that I can not take off it. Last，your sentence 学生总是不安静 is also right，but the tone is not that strong，if you say 学生总是安静不下来，meaning it is very hard even impossible for them to keep silent. Hope this can help you. 希望这个可以帮到你。

“”我的手机在柜子上。 请帮我吧手机搬下来。“”这里不能用搬，手机不大不需要用搬这个词，应该用带下来。

"你知道，学生总是不安静"——这句话描述的是一种状态，意思是学生们总是处在吵吵嚷嚷的状态中。 "你知道，学生总是安静不下来"——这句话描述的是一个过程，意思是学生们总是无法从吵吵嚷嚷的状态逐渐进入到安安静静的状态。 从日常用语角度来讲，"你知道，学生总是安静不下来"和“你知道，学生总是闹哄哄的”更自然一些。 “我的手机在柜子上。 请帮我吧手机搬下来。”的正确说法是“我的手机在柜子上，请帮我把它拿下来。”

"你知道，学生总是安静不下来" 安静 = 动词 verb 不下来 = usually it can use after the verb to describe that action is continuously or unable to have a complete action. E.g. 1 安静不下来 = cannot keep it quietly 我的手机在柜子上。 请帮我把手机搬下来。(are these sentence correct?) 搬 = usually used for carry something is very heavy, so handphone is very light weight and it not suitable to use 搬 Correct = “我的手机在柜子上，请帮我把它 (it, mean the 手机) 拿下来。” +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ "你知道，学生总是不安静" - [It could be better if write it in this way - "你知道，学生总是吵吵闹闹 （chǎo chao nào nao）的"

The difference is a bit subtle. I'll give maybe not a word-by-word translation and a bit of context. 学生总是不安静 - Kids, in most cases, aren't very quiet. Maybe the speaker is describing to someone how things usually are with the kids. 学生总是安静不下来 - Kids just won't quiet down. Maybe the speaker has been trying for 30 minutes to get the kids to be quiet and fails to do so, and then says this. This first is a picture, a state, or a habit. The second is emphasizing on the PROCESS of getting into that state, or the failure thereof. The part '下来' is quite similar to adding 'down' after 'quite': 冷静 vs 冷静下来 - calm vs calm DOWN