Are these statements true or false (Mathematics) ?

Translate the statements into English.

1. For every real number x and every natural number n, there exists a real number y that is greater than x+n.

Well, x+n is a real number, so y=x+n+1 is a real number greater than x+n. True.

2. For every real number x, there exists a natural number n such that for every real number y, the quantity x^2 + n^2 is less than or equal to y.

The quantity x^2 + n^2 is a real number, and that can't be less than or equal to every real number y. False.

Try this type of thinking on #3. (Note that the earlier answer's suggestion of "solve for the 'exists' variable" doesn't work on this one.)

That notation you're being taught is easy to write, but doesn't make "order of operations" very clear. I like the notation I was taught in symbolic logic that adds parentheses and brackets:
(∃ x ∈ Q) [ ...statement about x... ]

The full version of #3 in this style is messier, but makes the "nesting" explicit:

(∃ x ∈ Q)[ (∀ n ∈ N)[ (∀ y ∈ R) [ y = n + x ] ] ]

Order is important. ∃ x ∀ y : x=y is not the same as ∀ x ∃ y: x=y, so you can't just "solve for the exists".

solve the equation or inequality for the variable quantified by ∃.
for all possible choices of the other variables, is there a solution?

And how do I determine this?