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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 $p_1$.