“Set”, as a verb (as in the first sentence of “D.D. Trap Hole”), means placing a monster face-down. So you can activate “D.D. Trap Hole” when your opponent Sets a monster from the hand, or Special Summons a monster face-down with “The Shallow Grave”, or flips his monster face-down with a card effect.
“Set”, as an adjective (as in the second sentence of “D.D. Trap Hole”), means a face-down monster. So if “Ceasefire” is chained to “D.D. Trap Hole”, the effect of “D.D. Trap Hole” disappears because the opponent’s monster is no longer Set.
If either targeted monster is removed from play before “D.D. Trap Hole” resolves, then the effect of “D.D. Trap Hole” disappears.
You cannot activate “D.D. Trap Hole” during the damage step.
If your opponent Sets more than 1 monster simultaneously, as with “Cyber Jar”, you cannot activate “D.D. Trap Hole”. You can only activate it when your opponent Sets 1 (and only 1) monster. (In the case of “The Shallow Grave” your opponent is only Setting 1 monster, even though you are Setting 1 too. So you can activate “D.D. Trap Hole”.)
You can select a Token as your monster to destroy and remove from play from your side of the field for “D.D. Trap Hole”.
“D.D. Trap Hole” is an effect that targets. It targets 1 of your monsters and your opponent’s monster that was just Set.
If you select your monster equipped with “Heavy Mech Support Platform” for your monster to destroy with “D.D. Trap Hole”, then “Heavy Mech Support Platform” is destroyed instead and is sent to the Graveyard. Your opponent’s monster is destroyed and removed from play.