Regarding appropriate usage and correct grammar, I don't think — and the comments support this— that the resources you've taken the sentences from may be called reliable.
At the same time, all the sentences are quite understandable and can be transformed into something more readable. But even leaving them as they are, you can't avoid "of" in every one of them:
- It looks like your teacher might have made a little of a mistake.
In this one, the answerer on the Yahoo page might mean to use the expression "to make little of something", meaning "to minimize or belittle something".
2.It was the last I ever saw of him.
"To see of someone" is a set phrase meaning "to meet someone; be in contact with him/her"
- Let go of an opportunity.
This really sounds weird since "to let go of something" means "to release, as from one's grip" and it's a set phrase too.
- I've kicked enough of these guys' butts.
You can't make do without the preposition of here as it expresses to whom the kicked butts belong.
5.It was an end [of] befitting of a Shinobi.
To me, this sounds even weirder than "let go of an opportunity", but even so, the preposition "of" asks to be inserted for it might indicate the connection between whatever a Shinobi is and being no longer suitable to or proper for someone or something (see the meaning of the verb "befit"). In other words, it might answer the question - not an iota less weird than the sentence itself - *"Whose end of befitting it was?"
6.He robbed me of my childhood.
"To rob someone of something" is a set phrase and it means "to deprive someone of something, not necessarily by theft".