12 string neck for TSRTS, 6 string neck for Rain Song. Me and my buddy kinda assumed that's why they didn't segue into into it, but then wondered how they did it in the old days. First viewing of Celebration Day gave us the answer.
Per the op, I think the Les Paul and the downtuning added alot to Kashmir, just really crushing. No idea which guitar he's used on it through the years, but I've always liked the heavier versions.
I was surprised to see him use the Transperformance gizmo to switch to an open tuning during Whole Lotta Love. Cool touch though.