Normalisation of wavefunction given by the form $Ae^{i(kx-wt)}$

Question 1.

Let’s say that the wavefunction is given in the form

$$Psi(x, t) = Ae^{i(kx-wt)}$$

Then because of the normalisation condition, the following should hold.

$$int Psi^*Psi dx = A^2 int_{-infty}^{infty} e^{-i(kx-wt)}times e^{i(kx-wt)} dx = 1$$

Because $e^{-i(kx-wt)}times e^{i(kx-wt)} = e^{-i(kx-wt) + i(kx-wt)} = 1$, the condition demands that

$$A^2 int_{-infty}^{infty} dx = 1$$

As the integral value diverges to $+infty$, we reach the conclusion that $A$ should converge to zero.

What’s wrong here?

Question 2.

This is another question that should be classified and asked separately but as it is a short one I will just put this one into here. When expressing the wavefunction as a linear combination of basis functions, especially in discrete cases, is it that the index varies from $-infty$ to $infty$? That means, is it that

$$Psi(x) = sum_{-infty}^{infty} c_i psi_i ?$$

Apologies in advance if the questions are trivial. I am a newcomer to quantum mechanics.