should have parameterization

if it's supposed to capture the sphere of radius
centered at the origin. (
is missing from the second component)
a. You should substitute
(missing
this time...). Then





as required.
b. We have




c. The surface area of
is

You don't need a substitution to compute this. The integration limits are constant, so you can separate the variables to get two integrals. You'd end up with

# # #
Looks like there's an altogether different question being asked now. Parameterize
by

with
and
. Then

The surface area of
is

The integrand doesn't depend on
, so integration with respect to
contributes a factor of
. Substitute
to get
. Then

# # #
Looks like yet another different question. No figure was included in your post, so I'll assume the normal vector points outward from the surface, away from the origin.
Parameterize
by

with
and
. Take the normal vector to
to be

Then the flux of
across
is



If instead the direction is toward the origin, the flux would be positive.