I flew with one of my girls 2 times, round trip, from California to New York.  Yes, you have to pay extra...it was about 4 years ago, and I had to pay $50 to have her stay with me.  Yes, they are supposed to be in a crate type thing that will fit under your seat (they will tell you the size). I had her in a soft crate, that zipped all around and had plenty of screening for air circulation. You definitely have to keep her in there when you're landing or taking off, but during the flight, if it's not too crowded, and depending on the stewardesses how nice they are, I kept her on my seat with me most of the time.  Try to sit in the back of the plane, or where the exits are, so you have more room.  And, if you take a nite flight, they are usually more lax.  I also had the vet give me some pills to take with me so I caould keep her sedated.  But, looking back on it, I didn't like giving her the drugs, it made her so dopey.  I think the movement of the airplane itself would've lulled her to sleep anyway.  Just don't be real obvious with her and it should work out fine.  Oh, and in the airport terminal, you have to keep them in the crate, too. But, when I was waiting for the plane, I took her out once again. 
