The fact that my neighbor, say, John is a horrible person does not imply that all people are horrible. Then a large part of life would lose meaning.

Both the premise and the conclusion of can be false in a particular interpretation, so p(a) can be true and q(a) can be false in that interpretation.

You also need to show that the sentence is invalid, i.e., you need to find a counter-interpretation.

I am not sure why you could not see it. In fact, I am not sure how changing into helps in seeing that the original sentence is satisfiable. The problem with your post before "EDIT," as I pointed out above, is that p(a) can be true and can still be false.

Edit: Additional remark. I don't see how B can be true for any sentence except when one considers validity and satisfiability with respect to an empty class of interpretations. If one considers all interpretations and a sentence is valid, then it is true in all interpretations, so it can't be unsatisfiable. Similarly, "satisfiable" is unnecessary in A since it is implied by "valid."