In (∀y)(∃x)Q(x,y), you choose x according to a given y, whereas in (∃x)(∀y)Q(x,y) you can't change x once you fix it. For example, when Q(x,y) means "x is wiser than y", (∀y)(∃x)Q(x,y) means that any y can be beaten by some x, whereas (∃x)(∀y)Q(x,y) means that there is some kind of god x(the wisest one) who is superior to every y. Try to interpret this into some mathematical example. To discover the flaw, interpret each line of the "proof " to know which line fails.
(∃x) Q(x) (and hence (∀x)P(x) V (∃x) Q(x)) happens even if Q is something rare, occuring for only one instance say, a. So, Q(a) occurs but Q(x) is false for all x except the case x=a. Then, it is obvious that you can't expect [P(x) V Q(x)] to occur for all x (since nothing is assumed for x except for the case x=a.)
There are many ways to disprove a wff (showing counterexamples, or using tableaux,...). I don't know which ones are used in your book.