$\sim (p\vee q)\vee (\sim p\wedge q)=(\sim p\wedge \sim q)\vee (\sim p\wedge q)=(\sim p\wedge (q\vee \sim q\left)\right)=\sim p$ 

$p\Rightarrow q$ is false only when p is true and q is false.
$p\Rightarrow \sim q$ is false when p is true and ~q is false i.e. when both p and q are true.

It can be seen that $\sim q\vee \sim p$ is false only when p and q are both true. Hence it is the equivalent of $p\Rightarrow q$

p: the earth is round
q: 3+4 =7
~p: It is not that the earth is round
~q: It is not that 3+4 =7
$\sim p\vee \sim q:$ It is not that the earth is round or it is not that 3+4 =7

$\sim (p\wedge q)=\sim p\wedge \sim q$
Hence option B is false.

P: Ravi races
Q: Ravi wins
~Q: Ravi does not win
$P\vee \sim Q:$ Ravi races or Ravi does not win
$\sim (P\vee \sim Q):$ It is not true that Ravi races or that Ravi does not win

$\sim \left(\right(p\vee \sim q)\wedge q)=\sim (p\vee \sim q)\vee \sim q=(\sim p\wedge q)\vee \sim q$

The negation of the statement would be 'not all cats scratch' or there exists at a cat that does not scratch.

'There exists' is the existential quantifier and 'for all' is the universal quantifier

The negation of $p\Rightarrow qisp\wedge \sim q$ 
p: We control population
q: We prosper
$p\wedge \sim q:$ We control population and we do not prosper

$\sim (p\Rightarrow q)$ is true only when p is true and q is false. Hence it is logically equivalent to $p\wedge \sim q$