There are pet courier services that will handle all the details including delivering the animal to you. Or you can arrange for a pet shipper to have the animal shipped through one of the major airlines, but you will have to pick the animal up at the airport. Look at this site www.nwa.com/services/shipping/cargo/products/ppet.shtml
Yes there are some air travel companies that are providing this facility to their customers for transportation of animals from one place to another by charging their fees. They can safely shift your cat to your desired place. This is not much expensive and it is feasible as well now.
Yes you can. The cat will be in with the baggage, which is not a problem. Do make sure that you have a secure cage with a locking door. If it's a long flight the attendants will give the cat food and/or water and you don't want the cat to leap out and get lost.