A Locus on a Cube's Surface

A cube $ABCD\text{-}A_1B_1C_1D_1$ has edge length $1$. A point $P$ moves over the surface of the cube with $PA = r$ (where $0 < r < \sqrt{3}$), and $f(r)$ denotes the total length of the locus of $P$.

1. Find $f\left(\dfrac{1}{2}\right)$. 2. Determine all possible values of the number of solutions of the equation $f(r) = k$ as $k$ ranges over the reals, with justification.