probability – question about the plan of the region when we transform pairs of random variables

Leave $ X $ Y $ Y $ They will be two independent random variables.

Leave $ Z = g (X, Y) $ A function of two random variables.
To find the CDF of $ Z $ we have

begin {align}
P {Z leq z } = & P {g (X, Y) leq z } \
= & iint_ {g (x, y) leq z} ^ {} f_X (x) f_Y (y) dxdy.
end {align}

In the books, Inequality. $ g (x, y) ≤ z $ defines a region in the $ (x, y) $ airplane.
How do we find this region? For example $ Z = X + Y $.

Another question is: Why do we move in capital letters? $ (g (X, Y)) $ a lowercase $ (g (x, y)) $?

Also for example the probability that a random variable $ X $ It is less then or equal to another random variable. $ Y $ one is
begin {align}
P {X leq Y } = & int_ {y = – infty} ^ {+ infty} int_ {x = – infty} ^ {y} f_X (x) f_Y (y) dxdy.
end {align}

Because it is not
begin {align}
P {X leq Y } = & int_ {y = – infty} ^ {+ infty} int_ {x = – infty} ^ {Y} f_X (x) f_Y (y) dxdy.
end {align}

I mean
$$
int_ {x = – infty} ^ {Y}
$$

do not

$$
int_ {x = – infty} ^ {y}.
$$