I have a room directly above a stronghold. I put wood doors on all openings to prevent mobs from getting in. But, I have had several instances of zombies coming right through the doors without the doors even opening!
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Yes, but the doors remain closed and intact! That just doesn't make any sense.

Please attach screenshots so that we can better understand what you are reporting.

I don't think a screenshot is necessary. The wooden door looks exactly like it did before the zombie came through it. You cannot see it open or close.
The zombie is just suddenly inside the room and attacking. You can first see it right by the door and so I am presuming it came through the door instead of spawning inside the room, which is well lit by torches.

Is there a Zombie spawner right below / besides the room ? Monster spawner spawn monsters through walls in a 9x9 area around the spawner.

Dupe of MC-2310
Zombies will sometimes break doors on Hard difficulty:
http://minecraft.gamepedia.com/Zombie#Attacking_villagers
To prevent this, use iron doors instead.