As long as the world is open (i.e. running on a server), the spawn chunks will remain loaded, whether or not anybody is actually in them.
As for mob spawning, I am yet to find any hard evidence as to whether or not any mobs naturally spawn in the spawn chunks, but normal events involving mobs can still happen:
- Passive mobs, such as baby animals and villagers, can grow into adults.
- Sheep can eat grass and grow wool.
However, due to the spawn chunks never unloading, any mobs in the spawn chunks will not de-spawn. This could possibly mean that mobs will not spawn naturally in the spawn chunks:
The passive mob spawn cap is limited by the number of friendly mobs loaded into memory, which means that any passive mobs present in the spawn chunks count towards the mob cap and will usually prevent friendly mobs from naturally spawning anywhere else in the world.