Hint for 1:
the vector parallel to the vector a is a constant times the vector a. a vector perpendicular to a is one for which its dot product with a is zero
you want these two vectors to add and give you b, you will have to solve for some unknowns
Hint for 2: if the plane is parallel, it has the same normal vector, namely <1,3,-5>.
a plane with normal vector $\displaystyle \vec n = <a,b,c>$ passing through a point $\displaystyle (x_0,y_0,z_0)$ is given by
$\displaystyle a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
the distance, $\displaystyle D$, from a point $\displaystyle (x_0,y_0,z_0)$ to a plane $\displaystyle ax + bx + cz + d = 0$ is given by:
$\displaystyle D = \frac {ax_0 + by_0 + cz_0 + d}{\sqrt{a^2 + b^2 + c^2}}$
Hint for 3: find c, let $\displaystyle c = <c_1,c_2,c_3>$ and take its dot product with b and find the vector when you subtract it from a, set the two vectors equal and solve for the components
Hint for 4: do the computation and set it equal zero. you should be able to see if this describes a line or not.
then set the same computation to a nonzero constant, show that you can't get a line
Hint for 5: come on, finding determinants is something you can look up, quite easily. google is your friend.
Hint for 6: again, something you can look up.