首先非谓语动词包括：不定式、动名词、分词(现在、过去分词).
1\ His left eye injured, he failed to notice the instruction of leaving the front line.（leaving,动名词）
2\ I'm happy to know that he is happy. (to know,不定式)
3\ I'm sure of my fellow‘s taking the microscope to the office last Saturday. (这里你要是较真的话,用company,但office更口语一些). 这句话非要用非谓语动词,写出来还挺别扭的.
4\ Please put those cloth hanging over the ceiling and eggs laid yesterday to the south-facing room.（这个句子里很多非谓语啊,hanging, laid. 是分词）
5\ When I went to the classroom to ask him for experiment suggestion, he happened to make dumplings by his clumsy hands in the kitchen.（你的中文前半句写错了吧,后面说碰巧在厨房,前面应该是在家才说得通呢）（to ask 不定式）
6\ The dam finished last year can ply the new plant with sufficient power. （前面finished 是分词作非谓语）