You're not doing anything wrong - just not going far enough.
(1) You have
If you write it as
. It's Bernoulli in .
(2) You have
which integrates giving .
This also integrates.
As for your first question, it doesn't matter. Define
then drop the bar.
For your second question, let go back when we introduced new coordiantes such that
gave your PDE by choosing
with the IC as when From the second two we find
which integrates gives . Imposing the IC gives giving .