Regarding your second question, the requirement that the inner product space be complete is imposed to have available a number of important theorems, among which the spectral theorem is particularly important.
After a bit of time, though, you might notice that there's a couple of fishy things in that Hilbert space axiom. For one, we put a lot of stock in position and momentum eigenstates, i.e. delta-function and plane-wave wavefunctions, which are strictly speaking not inside the Hilbert space. On the other hand, the normal Hilbert spaces have a number of wavefunctions, such as $$\psi(x)=\frac{1}{\sqrt{\pi}}\frac{1}{\sqrt{1+x^2}}$$ which violate physical intuition in one way or another (this one has infinite position dispersion). The resolution is an amendment to the Hilbert space axiom in terms of rigged Hilbert spaces.
