Most, if not all zeds, and some instances of wildlife, can bash doors down. Closing doors will slow
them, sometimes immensely, but never completely stop them unless the doors are metal. Even reinforced wooden doors can be bashed down, albeit more slowly.
As for the apparent teleportation, it’s most likely the zed following you up or down the stairs. Creatures can follow you up or down them, and don’t need to be on the stairs to do it; they just need to be near them. If they are within 2-or-3 squares in all directions from a set of stairs that you go up or down, they will automatically be pulled with you after a few turns, usually indicated in the message log with “%Creature% is coming down the stairs”, and “%Creature% is almost at the bottom/top of the stairs”. All monsters within the radius will be pulled down, even if they are busy with other things, such as munching on your NPC. The game pulls them down if they are in the radius, even if the creatures don’t see you, so think of it as if they were close enough to hear you going down the steps, and thus followed the noise.
If you’re on the same step that they will spawn onto, it’s also possible for them to push you to the side when they do spawn. The radius for creatures to follow you up stairs may also ignore interior obstructions such as doors, but I doubt that’s the case.
When you slept, the spitter was most likely within the range to follow you down the steps, but you either didn’t notice the message, or you had already closed the door and thus broken the LoS, so the character couldn’t “see” the zed coming down the steps. Then it bashed the door down while time sped up because you slept, and got to you. You went back up the steps, and it probably followed close enough to get back up them as well. If it was a spitter, then you were probably in considerable pain, and thus slowed.
