When I was returning from my interview in Vermont I flew out of Burlington, a tiny airport. I flew direct from Burlington to Chicago. My luggage went from Burlington to Washington D.C., to Chicago a full day later. Go figure. At least my luggage wasn't lost on the way TO the interview. Now, you might wonder why I didn't take my suit/luggage as carry on. That's because I took my dog!
So, yes, you can take a cat or dog on the plane with you. It must be in an approved under the seat carrier. I've taken my dog on American and United. They've never required any proof of health or vaccinations to fly (although he's got them). It's generally $80 each way, and you then can't bring a carry-on. Just the carrier and a purse (or equivalent). And remember to take your pet out of the carrier before you go through the xray machine.
I've never done it, but I know many people who have - with meds given to them by their vets.
Anyway, my dog is silent as a mouse. He's been on about eight round-trips now (including my GW interview), and no one has ever even realized I had a dog with me unless I specifically told them (I always tell the person next to me in case they're allergic). He just sleeps or watches me in silence. So, not everyone who brings their dog is that inconsiderate. And I have a rat terrier, but he is by no means a "rat dog." A respectable 25 lbs. And we had a great time in Burlington and D.C. Certainly more fun than taking a trip alone.