mathematical physics - State normalization in Dirac's formulation of quantum mechanics

Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our states are normalized by $\langle q' | q \rangle = δ(q' − q)$ with $δ$ the Dirac delta function. ("Dirac's formulation, in the book "Quantum Field Theory" by A. Zee)

1) Why must $\langle q' | q \rangle = δ(q' − q)$ hold?

2) Is this related to $\int_{-\infty}^{\infty}\Psi^*\Psi dx = 1$?

Also,

Now use the fact that $|q \rangle$ forms a complete set of states so that $\int dq |q \rangle \langle q| = 1$.

3) Does this even make sense?

4) Shouldn't integral be $\int_{-\infty}^{\infty}$?