"Its my opponents turn. I have MRS set facedown. He uses MST and targets MRS. I activate MRS. It resolves first, goes to a monster slot (right?) in defense mode. Does MST destroy MRS? The MT spot that MST targeted is clear."
This does not work like you're thinking. Metal Reflect Slime is considered in your Monster Card Zone AND the Magic & Trap Card Zone you Set it in when it is Special Summoned. So even if you chain it in response to Mystical Space Typhoon, Heavy Storm, Giant Trunade, etc, it is also still in your Magic & Trap Card Zone, but "connected" to your Monster Card Zone by the fact that it is still treated as a Trap Card.
"Second scenario: If I have MRS and Skill Drain faceup, does Skill Drain negate the would-be monster effects (eg. This card is still treated as a Trap card)?"
Based on the ruling made for Dead Dragon - Zoma/"Zoma the Spirit"
"Divine Wrath" cannot be chained to "Zoma the Spirit's" effect because it is a Trap Card in the Graveyard. "Sorcerer of Dark Magic's" effect cannot be chained to the damage inflicting effect of "Zoma the Spirit". "Zoma the Spirit's" effect cannot be negated by "Skill Drain".