1. The brain probably chunks this cluase to "once," thereby creating one chunk, instead of two, in STM. (See our psycholinguistic model of reading.) However, if you chunked it as an adverb to "began running," count your answer as also correct.

2. Although this semicolon does separate main clauses, I don't see any contrast between the ideas expressed in them. I note this to suggest that many of our "rules" are simply norms -- which I would enforce upon students with caution. See semicolons.

3. You could consider "cried the little mouse" as a subordinate clause functioning as an intejection within "Pardon, O King, forgive me this time."

4. Technically, this is a comma-splice, but the main clauses are short.

5. I will guarantee that you will get at least five different answers from a dozen different grammarians in explanation of this "but what." If students are struggling to master clauses, this is not worth spending time on. I would slide right over it unless someone specifically asked about it. In that case, the "but" may be explained through ellipsis -- "who knows *anything* but what I may be able ...." I would then consider the "but" as a preposition (It means "except."), and the clause as the object of the preposition. This still leaves a problem with "what," which, in terms of my explanation, should be "that." I have a sense, however, that  "who knows but what," which means "perhaps," is what Roy O'Donnell would have considered a "formula," -- a series of words learned as a whole without regard to its inner syntactic structure. As I stated above, this type of question can take a lot of time, and a lot of research.

6. I'm wondering how many readers are momentarily confused by this "after." I first read it as a subordinate conjunction ["after the Lion was caught in a trap ..."] I then hit the "and," crashed, and had to reprocess the sentence. The problem would not occur if there were a "that" after "after": "Some time after that the Lion was caught in a trap, and ...."