According to the Minecraft Wiki, aggressive mobs won't wake you if you surround your bed with torches. Here is what it says about the mob behavior:
To ensure that you are not awakened unnecessarily during the night, the bed must not be placed along a 1 block thick wall, unless the area on the other side of the wall is also safe. If a hostile mob can get within 2 blocks of the bed, even if the blocks are separated by a wall, you will be awakened with the hostile mob standing next to you. There appears to be a slight delay before the mob attacks you, giving you time to get the first blow.
Edit: As of Minecraft 1.0, mobs will not awaken you suddenly if your bed is placed in an unideal location. Instead, the game won't let you sleep and will state that there are mobs nearby.
This question and answer apply to Minecraft 1.4.7 and earlier. Minecraft 1.5 and later support high-resolution textures automatically, and do not exhibit the misplaced animations described below.
Minecraft versions before 1.5 were not written to support different-resolution textures.
The default Minecraft terrain texture in these versions is a 256×256 image with 16×16 tiles in it (a texture atlas).
The code which animates the animated textures in Minecraft (fire, water, lava, and portals) writes onto specific pixel offsets according to 16×16 tiles, which are wrong for larger textures, so you get small fire/water/portal images occupying other blocks. Presumably the patchers replace the 16× factors with 32×, or whatever fits the actual texture resolution.
The reason the textures mostly work is that the OpenGL 3D API used by Minecraft mostly takes coordinates in textures in terms of numbers between 0 and 1, rather than 0 and whatever the pixel size of the image is, so the main rendering doesn't care; only the animated textures, which are actually defined in terms of recalculating individual pixels and modifying the data of the texture in memory, are affected.
Also, tiles over a certain size (128×128, I think) will crash Minecraft. Below that size, you can still use HD textures and it will work except for the misplaced animations. The animated items will not animate, and (for a 32× pack) these will have animations on them: bricks, gold blocks, Netherrack, and pumpkins.
Best Answer
It looks like a resource pack or mod has been installed. If it's a resource pack, then you should be able to go into the resource pack list and turn it off. If it's a mod, I wouldn't have the slightest idea how to uninstall it, but there are probably multiple tutorials on the Internet willing to tell you.
About that resource pack you might think isn't doing it, try turning it off and going to sleep. If it changes back, keep it off. If it doesn't try looking for mods that might be causing this problem.