I don't get what the problem is here, last time (31.25) I tried a dwarven bathtub, I basically just dug a 1 tile wide channel on my main hallway, had it fill up to 3/7 with water using bucket brigade and it worked like a charm.
On the outside there was an evil rain, my dwarves would get covered in the muck, get inside, pass through the flooded channel and they would drop all the muck on the channel and get a coating of water all over, even when the water was already contaminated they would still leave with a water covering, so where's the problem here?
Animals. Well, mostly animals, a few serfs without proper clothing here and there. The bathtub works great if the creature stepping on the ramp has shoes or boots (socks are probably just as good), but the insect and vermin killers will catch any and all contagion via their feet, and then carry it into the fort. When they die you get tantrums (and I'm not sure, but can't some syndrome spread from the dead host it infected?)
That's usually how I first realize I'm on the leading edge of a syndrome outbreak; cats start exploding, guard dogs start keeling over, etc. Don't get me wrong, I'm alright seeing casualties on the battlefield and even sickened or dead from FB goo while confronting it, but syndrome has some seriously latent effects sometimes. I've made it my #1 nemesis.
Socks are good enough, I almost never makes shoes since I runs civilian military. Though animals, can't really do much with them, unfortunately, outside of pasturing them or covering the goop bath with grates/hatches to move animals to other side.
Syndromes don't technically spread, in sense of being a disease, they have to be breathed in, smeared on skin or injected into blood. Contact syndrome's the source of most reports since the frigging thing gets everywhere and sticks around for loooong time. Some of them can have hugely delayed effect, too. Splatters can multiply if it's left around, too, so that don't helps for the contact ones.
Well, come to think of it, contact syndromes are like disease, huh? Well, one that can be prevented by covering up, but still.
My usual procedure is to use hatches or grates to clean up non-bath splatters, I think roads and bridges works, too, but they takes too long for me and dwarves are way too slow to clean up on their own. Civilian militia usually have plenty of boots by time forgotten beasts shows up, anyone with hauling enabled pretty much must have boots and gauntlets for me, that's probably why I don't really experience much contact issues. Helps that only animals I really keeps around are the birds for their eggs, and they're stashed out of the way.