Minecraft – Why are there flying pigs in the Minecraft

The conditions shown in your screenshot should be able to spawn pigs, assuming it's a pig spawner. I did a rough recreation of your setup in my world.

Even in the dark, pigs were spawning, and within seconds of placing the spawner, so it's not as if you aren't waiting long enough. (Yes, the pigs also spawned when I surrounded the spawner with Glowstone). Because of this, I have a feeling that the spawner in your image isn't actually a pig spawner; it's likely a zombie spawner, or some other hostile mob. This is especially likely if you found the spawner and it wasn't manually placed by anyone, since there wouldn't be any natural pig spawners around.

Like Kevin Reid said, you could test this by making the room dark and seeing if any hostile mobs spawn. If you have admin privileges (and don't feel like this is cheating in some sort of way), you could use /give 52 1 to give yourself an actual pig spawner, place it in the room, and see if that one spawns pigs, just to verify if this is an odd SMP glitch that is only happening for you.

Mobs cannot spawn in a well lit area, but only a single square is enough to allow spawning. The light level must be 8 or higher to prevent spawning, meaning no cube can be more than 7 steps from a torch. You have to count vertical steps too, so a torch two squares off the floor can only be five horizontal steps from any floor square.

In addition, spiders can climb up walls, and thus are harder to keep out than other mobs. But the fact that you have creepers and zombies as well indicates that you have a lighting issue. Creeps can also fall down from heights farther than they would choose to walk; if the cliff above your balcony can spawn mobs, you may wish to light it or fence it to prevent mobs from pushing each other off the edge and falling to your balcony.

After you posted the screenshots, I made this graphic, illustrating some of the squares that have insufficient light. The green dot means that area is light level 8 or higher (within 7 cubes of a torch). A red dot indicates it is light level 7 or lower, and mobs can spawn there. As you can see, there is a large splotch of darkness on the right side of your balcony, where it is widest. Add a few more torches and everything should be fine. I did miscount a bit when creating the graphic: the leftmost two red dots should be green, as they are 7 cubes from the torch that is just out of sight on the bottom left. This leaves 5 squares visible in the image that are below light level 8.