I'll talk here about the last three problems from the How Logical Are You? quiz.
Question 6: One of the very tricky things about logic problems with all/none/some problems is that English is ambiguous and inconsistent about what "not", "none" or "no" really apply to. If the problem doesn't use "some" (see question 8), we can use material implications. "All P's are Q's" means "if P then Q" (p -> q). "No P's are Q's" means "if not P then not Q" as well as "if not Q then not P" (~p -> ~q and ~q -> ~p). Again, we can extract the valid statements by using the contrapositive (see question 1).
If you're middle-aged as I am, you might remember Venn diagrams from your New Math class. A Venn diagram is a good way of visualizing all/some/none relationships.
Notice that "All A's are B's" is asymmetrical: A's relationship to B is different from B's relationship to A. That's why we use the asymmetrical if ... then... On the other hand "No A's are B's" is symmetrical: A has the same relationship to B as B does to A. That's why we use two material implications.
One important trick in logic questions is that "not true" does not always mean "definitely false" (and "not false" does not always mean "definitely true"). In many cases some assertions are—given the premises—uncertain, not determinate, unknown. You can validly assert neither their truth or falsity. This question tries to trick you because it specifies a relationship between musicians and chefs and a relationship between teachers and chefs, but it doesn't specify any relationship at all between musicians and teachers.
Question 7: This question is not a logic question per se, but rather a pattern recognition question. There's a general pattern in these types of problems: The patterns that come earlier in the test are usually simple and direct; they become more complicated and abstract as the test progresses. Since this is the first pattern-type question, the pattern you're looking for is indeed fairly simple.
Once you get the pattern, you can get the answer with some simple arithmetic.
Question 8: Like question 6, this is a all/none/some question, but this question uses "some". "Some A's are B's" (and "some A's are not B's"), however, gives us very little information, so little that you can rarely draw any sort of valid conclusions. The Venn diagrams are ambiguous:
Since "some" questions are ambiguous, you want to look for "none of the above". The definitely true or false statements you can extract from a "some" condition are so grammatically complex and ambiguous that test makers rarely bother to include them.