I went mining in minecraft and came back home to a skeleton inside, my doors all shut. Can skeletons open doors?

Yes

Monsters will spawn if there isn't enough light. If you have no torches or not enough torches in your house then monsters may spawn inside.

To kill an enderman you need to kill it with a sword. Aim for their feet as they're likely to teleport if you aim higher.

The room might be too dark, try adding more torches/glowstone to light it up. The skeleton could have just spawned in there

Skeleton's can't open doors. On the other hand, if there's anywhere in your house dark enough for monsters to spawn, skeleton's could just appear inside your house.

Endermen can be kill with any *melee* weapon, or even your firsts if you want. All you need to do is hit it enough. Endermen are immune to ranged weapons, they always teleport out of the way. (You can also just walk away. If you move far enough away from a monster the game will delete it.)

House may be too dark so skeleton may spawn and no they can't open doors. Enderman can be killed by any melee attack don't uses bows cause they won't work