Hi, could you check these sentences, please?
1) I insist that his dog <s>back off from</s> get off of my lawn immediately. [Note: you can also tell someone to back off, but that's more often a general request to be left alone rather than a demand that they get away from a specific place. You could also say 'back up off of my lawn', but that's very colloquial.]
2) If you didn't back me up I would<s> have big troubles (</s>either: 'would have been in big trouble' or 'would have had seriousl problems'). So I don't even know how I can express my gratitude.
3) I didn't see the level of diligence that I expected from him.
4) He's too fussy to live in nature in tents for several days in the row. (or should I use picky instead of fussy?) [Either is fine! Fussy sounds slightly better to me here, but matter of taste.]
5) I felt a strong exasperation [more natural: 'I felt really exasperated'] after coming home from work, so I asked that no one bother me till tomorrow.
6) One of his most valuable qualities was integrity.
7) I want to acquire new skills in order to become a better employee.
Great job!