Also worth noting that the caverns are particularly susceptible to undeadsplosions, where cavern creatures come into the caverns and immediately get killed by undead, which lets new cavern creatures come into the caverns and get immediately killed by undead, snowballing into a massive unstoppable lag machine (and fort destroyer if you try to open it up and defeat the zombie army).
Forgotten beasts are equally horrifying, yeah. They might take out some weaker zombies before dying, but when reanimated they're zombie insurance so that no individually powerful creature can wipe out all the zombies in that layer, since fbs are the cream of the crop. And inorganic fbs will just ignore the zombies and meander about killing (and subsequently reanimating) stuff.
Handling a poultry industry shouldn't be too hard. I haven't had any issue with the unkillable hair bug, and there's plenty of precautions you can take to mitigate or stop the effects. First, make sure the whole process gets done quick. Butcher, tan, and spin whatever you can ASAP, so make sure everybody who does those jobs is free before ordering stuff butchered. Second, train the dwarves who will be handling the work in mace or hammer (mace is actually better here, it's better at pulping even if it's a worse weapon) so that they instantly take out reanimated bits. Soldiers shouldn't be necessary (if you encounter the unkillable hair bug more soldiers will just make things worse, and if it's not unkillable then you don't need anything more than one dwarf to take it out), but can serve as an alternative. Third, place cage traps everywhere that potential reanimating bits will be (around workshops and anywhere the bits might be hauled across, but you should minimize hauling anyways). If you do get a buggy unkillable hair, do your best to evacuate the area and the cage traps will handle it. I've run a decades-long sheep industry in a reanimating biome this way with no incidents that weren't immediately resolved by the butcher instantly putting down the reanimated hair or skin, so it's quite safe.
The biggest potential problem is killing your poultry off incorrectly and letting one die of age while still in the pen. You'll only have a little bit of time to get the corpse out of there before it reanimates, so I might also suggest having two distinct enclosures for your poultry industry to prevent any incidents of this nature from destroying your entire industry. Two enclosures also just prevents it naturally: when you need meat, butcher one enclosure and repopulate with two poultry from the other one.