Do you "cower" everytime you roll doubles, even snake eyes, when shooting?
You cower, even on snake eyes, unless an SMC (like firing a MG), berserk, Fanatic, British elite and first line, Finnish, ordnance, OBA or vehicular.
Do you go under DM everytime you fail a MC or get Casualty reduced?
You go under DM when you break, even voluntarily.
You also go under DM if you are already broken and:
a) you're attacked by White Phosphorus
b) you're attacked with "enough FP to possibly inflict a NMC on the target" (even if the firing unit cowers). Ordnance must secure a hit to be considered an 'attack'
c) you're in a Location attacked by an effective (1 or 2 result) Sniper
d) a Known enemy unit becomes ADJACENT to you (even if the enemy unit Moves on, even if it is an enemy unit advance)
Casualty Reduction alone does not cause you to go DM. A77.302 (first edition) says "A squad is reduced to a HS with the same broken/non-broken status of the squad it was Reduced from.."