I do this when I fly musical instruments like my bass guitar (for which I haven't managed to purchase a hard-shell flight case, bah) to evade having a damaged instrument at my destination. Some airlines will let you store these things in the bulkhead compartment forward, and others require you to purchase an additional ticket.
Actually, no. I'd never flown with a dog before and was a rank amateur; it didn't occur to me that dog crates might be a different category of luggage/cargo. I freely admit that this was my fault (in a way), and I'm not saying they should have flown the dog anyway - but the way they handled it was not good. Instead of just saying, "I'm sorry, we can't fly dogs because they're a different category of luggage we're not permitted to handle here" and refunding my money, they were very strange about it and didn't refund my money.
So, yes, checking the airline's policies would've been top priority. With JetBlue, this is amazingly easy: http://help.jetblue.com/SRVS/CGI-BIN/webisapi.dll?New,Kb=ask...