I have a pretty bad dog allergy. It's bad enough that I get an attack by being near some dog owners. I wouldn't say it's life threatening though, but exposure to dogs can mess me up for days on end. Been a few times I probably should have gone to the urgent Care or ER for it.
No, I don't carry an epi-pen. My allergist recommends, but it's easier to just avoid exposure...that, and pre-treating with Allegra generally does the trick.
If I was on a plane with a dog, depending on my frame of mind I might pitch a fit too. There are few things that would freak me out more than being in an enclosed tube for hours on end at 36k feet, no Allegra, and a dog near by. Couple the allergy with anxiety, add lack of sleep, alcohol, and frustration with being treated like cattle, and you have a recipe for trouble.
If the dog stayed in a baggie or something, maybe that would be ok. But so many #######s out there think everyone wants to pet their dog and so they are quick to show off that mongrel...
I would add that while a dog allergy may not be bad on it's own, it can compound other things. For me, it's not the runny nose/itchy eyes, it's that it triggers my asthma to the point where my rescue inhaler (which I have carried in my pocket since 1983) doesn't give me relief. If I was recovering from the flu, or if my asthma was already teetering on uncontrolled (i.e. spring fever could do it), extended exposure to a dog could be a potential hospitalization event.