I heard if you change something called RequiredPlayerRange to a high number on a spawner, it will be able to spawn even from a distance. I don't know to change Tile Entities though. How exactly do I change that?
Minecraft – How to change Tile Entities
minecraft-java-edition
Related Solutions
As of the Minecraft version at the time of writing this...
Tamed wolves are not a passive mob and they do not have any limit, so they will not affect the passive mob limit (because if they were counted in the passive mob limit there would be a limit on the number you can tame.)
Passive mobs (a pig, for example) will NEVER attack you, even if you attack them.
Wolves are actually tame, meaning they follow and protect you.
Short answer:
Tamed wolves do NOT affect the passive mob limit in any way.
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.
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Best Answer
You will need to edit the NBTData of the spawner. Sethbling has this filter which can be downloaded onto MCEdit to change most spawner properties.
If you are not familiar with MCEdit or want a way to do it in game I would recommend this mod. You will need to look at the spawner, type /nbtedit, and then change RequiredPlayerRange.