This was classic for my Jack. He had several problems, but his liver
was bad. Being a Yorkie, it's normal for them to make a snort sound,
but I have not heard this for poodles before? He may have kennel cough?
Has she been around other dogs or cats lately? The can get this cough
and also allegeries like people. I would have her checked out by a vet
for sure.
Mark
You don't mention how old your dog is - though I suspect, given the bad teeth, not young. There are just so many things that could cause a dog to cough and gagg that it is impossible to suggest any particular course of action, other than to get her to a vet as early as you can. It could be a problem associated with the mouth, throat, stomach, intestines - or even something like heartworm or a blockage. It could also be tracheal collapse, which is a very common issue with this breed.
Toy poodles are particularly prone to heart disease, and the coughing and gagging could potentially be a symptom of there being a heart condition present. Again, a vet would be able to diagnose if this is the problem.
As to how long a mini-poodle lives, on average 10-14yrs, but some live up to 19yrs.
Tony