I find your answer puzzling.
Lets say we are dealing with $\displaystyle \mathbb{N}$ and $\displaystyle p(m,n)$ means $\displaystyle m\le n$.
Then $\displaystyle (\exists x)(\forall y)[p(x,y)]$ says "some natural number precedes every natural number". Does that imply that "every natural is preceded by some natural number"?
Can you use IE, UI, UG & EU in a valid way to get what you need?
Im not quite sure what you mean with your last question. But this is how i understand the sentence.
$\displaystyle \exists x \forall y p(x,y) $
"There exist an X where all Y is in the same function p(X,Y) which implies that for all Y is there an X in the same function p(X,Y)"
First of all i need to remove the imply arrow by negating the right side and split up the left and right part like this:
$\displaystyle \exists x \forall y p(x,y) , \neg ( \forall y \exists x p(x,y) ) $
You can then remove $\displaystyle (\exists x)$ by introducing a new constant for X.
$\displaystyle \forall y p(A,y) , \neg ( \forall y \exists x p(x,y) ) $