Find the negation of the following quantified sentence ( ∀ x ) ( p (...

Step 1
Notes: $\neg (\vee )=\wedge$
Not quite: the negation of $A\vee B$ is $\mathrm{\neg }A\wedge \mathrm{\neg }B.$.
Remember, you are negating a sentence, not merely its main logical connective.
$\mathrm{\neg }(\to )$
Not quite: the negation of $A\to B$ is $A\wedge \mathrm{\neg }B.$
$\mathrm{\neg }(\forall )$
Not quite: the negation of $\mathrm{\neg }\forall xA(x)$ is $\exists x\mathrm{\neg }A(x).$.
$\mathrm{\neg }(\neg q(x))$
Correct.
Step 2
So: $(\forall x)(p(x)\vee q(x)\to \neg q(x))$
Liberally inserting parentheses helps disambiguate sentences. In any case, the precedence of logical connectives dictates that this sentence ought to be read as $\forall x((p(x)\vee q(x))\to \neg q(x))$
(I removed the unnecessary parentheses around the quantifier).
Denying: $(\exists x)(p(x)\wedge q(x)\wedge q(x))$
This is incorrect, as pointed out in the comments and other answers. My above corrections make clear why.

Step 1
Using the definition of $P\to Q$, which is $(\mathrm{\neg }P)\vee Q$.
Step 2
So, since $\mathrm{\forall }x(p(x)\vee q(x)\to \mathrm{\neg }q(x)$ is the same as $\mathrm{\forall }x\mathrm{\neg }(p(x)\vee q(x))\vee \mathrm{\neg }q(x)$, its negation is $\mathrm{\exists }x\mathrm{\neg }\mathrm{\neg }(p(x)\vee q(x))\wedge \mathrm{\neg }\mathrm{\neg }q(x),\text{i.e.}\mathrm{\exists }x(p(x)\vee q(x))\wedge q(x).$