Let me address just Q2. The set of points at distance $D$ from a point $p_1$ in $C_1$ form an annulus
centered on the center of $C_1$,
of outer radius $D+r_1$ and inner radius $\max \{ D-r_1, 0 \}$.  So you just intersect this annulus with $C_2$,
obtaining (in general) zero, one, or two arcs of $C_2$ 
for the locus of $p_2$ points that are distance $D$
from some point $p_1$.