Not sure when to use Distributive Law in specific Proofs question.

Question

I'm currently using "How to Prove It" by Velleman, and I'm stuck on Chapter 2.2 question 6. I found the solution online on how to do it, but I don't understand it.

Here's the question:

Show that ∃x(P(x) V Q(x)) equals ∃xP(x) V ∃xQ(x).

Here's the answer:

¬¬∃x(P(x) V Q(x))=
¬∀x¬(P(x) V Q(x))=
¬∀x(¬P(x) ∧ ¬Q(x))=
¬(∀x¬P(x) ∧ ∀x¬Q(x))= (a)
¬∀x¬P(x) V ¬∀x¬Q(x) (b)
∃x¬¬P(x) ∨ ∃x¬¬Q(x)
∃xP(x) V ∃xQ(x)

My problem:

1. Why are we able to use the distributive law here when we couldn't use it to solve the whole question using it in the first place.
2. Here we used the negation law so that changed the middle sign. What I'm wondering is why the ¬ next to the P and Q is not removed, since we're supposed to reverse everything, including the sign too [like in $-[P(x)] = - P(x)]$.

Answer 1

This is more a comment than an direct answer to your question, but somehow I doubt that most mathematicians would be applying such rules (really theorems of logic) in writing proofs. While such rules may save a few lines of proof, it may be easier to simply apply the more basic rules of logic. Here are the first 5 of 28 lines in my formal proof (in the DC Proof format) to get you started. (Note: Here, '|' is the OR-operator.)