Understanding Wave-functions and Normalization?

Quelsita
Question:

1. An electron is freely moving in a one‐dimensional coordinate,
x . At some point t in time, its (complex‐valued) wavefunction
is ψ (x,t) = Ceiωte−(x / a)2 .

a. Why must $$\int$$ $$\left|$$$$\Psi$$|2=1?
b. From the so‐called normalization requirement given in
part a., determine the normalization constant C, assuming
it is a positive real number.
c. What is the unit of C in SI?
d. What is the probability density at the origin?

Basically right now I'm just trying to wrap my head around the concept of normalization etc but figured to list the whole problem to see where things were headed.
I get that the square of the absolute value of the wave function gives the probability density of finding a particle (in this case, an electron) in space.
And in Quantum Mechanics, all real particles must be "normalizable", meaning that the chance that said particle occupies some space must be 1. Is this just saying that for an electron in space, it has to be somewhere at sometime? That it simply can't NOT exist in space?
Also, must it only =1?

To be more specific, in one-dimension $$|\psi(x)|^2$$ gives the probability that the particle is found in the interval from $$x$$ to $$x+d$$.
If the quantum state is not such that the wavefunction is $$\psi(x)=0$$ for all $$x$$, then the probability of finding the particle somewhere cannot be zero. Said another way, the "no particle" state is the zero wavefunction.