Question

A narrow wire is used to slow the flow of the electrons in a circuit

Answer 1

That's true.

In fact, the resistance of a wire is given by:
$R= \frac{\rho L}{A}$
where
$\rho$ is the resistivity of the material
L is the length of the wire
A is the cross-sectional area of the wire

We see that the resistance of the wire is inversely proportional to the cross-sectional area: A. Therefore, the narrower the wire, the smaller A, the larger the resistance. But higher resistance means that the current flowing through the wire is lower, therefore the flow of electrons in the circuit is slower, and the initial sentence is true.