“Drillroid” can use its effect to destroy face-up and face-down Defense Position monsters that it attacks.
Note that “Drillroid” does not say “without flipping the monster face-up”. The effect of “Drillroid” is applied after the monster is flipped face-up, but before damage calculation. If the Defense Position monster was face-down and has a Flip Effect, the Flip Effect will activate and resolve normally.
The effect of “Drillroid” is a Trigger Effect that can be chained to.
If “Drillroid” attacks a face-up “Sasuke Samurai #4”, do the coin toss for “Sasuke Samurai #4” first, at the beginning of the Damage Step. If “Sasuke Samurai #4” destroys “Drillroid” with its effect, the effect of “Drillroid” is not applied. (If “Sasuke Samurai #4” was face-down, then you miss the chance to use its effect.)
If “Drillroid” attacks “Dreamsprite”, both are mandatory Trigger Effects with the same timing, so they form a chain with “Dreamsprite” as Chain Link 2. “Dreamsprite” resolves first and, according to her effect, causes damage calculation to happen immediately. Since the effect of “Drillroid” only happens without damage calculation, the effect of “Drillroid” disappears.
If “Drillroid” attacks a Defense Position “D.D. Warrior Lady”, the effect of “D.D. Warrior Lady” does not activate, because she has been destroyed by the effect of “Drillroid” by the time her effect would normally activate.
If “Drillroid” attacks a face-down Defense Position monster while “Final Attack Orders” is active, the monster is not destroyed because it flips face-up and immediately shifts to Attack Position.
“Nightmare Penguin” will activate its effect if attacked by “Drillroid” while face-down. (Just like a Flip Effect.)