former airline employee here. things may have changed a lil after the tsa took over the checkpoints, but this should still be valid. for any questions regarding what can and can't fly, print and take this list with you:
http://www.tsa.gov/public/interweb/a..._16_2003v2.pdf
now, the reason they don't like carseats:
people try hiding things in the wackiest places.
you bought two seats, so you can do whatever you damn want with that second seat, including putting a carseat into it. carseats are allowed through the checkpoint, but are subject to a search. if it dosen't fit through their x-ray machine, demand that they hand search it. don't let them push you around, you are within your rights in asking this.
as far as travelling, MAKE SURE YOU GET PREASSIGNED SEATS TOGETHER. call your airline now and ask them to do it.
if they can't give you a solid answer with seat numbers, then call the airline as early as possible on your day of departure and do it. seats that aren't available before will be on the day of departure. don't ask for exit row seating, you're not allowed to have kids there. if you aren't able to get seats together, get onto the plane as soon as possible (they should allow you to preboard because you're travelling with the munchkin), and ask the flight attendants and the passengers around you if anyone would be able to accomidate you. don't lose your temper or get hysterical, most people are more than happy to put a mother and child together (noone wants a screaming unatended child next to them).
during the flight, do your damndest to put that child to sleep. sleeping children are a blessing on any flight. if you can't keep the kid asleep, keep them occupied with something they won't get bored with. take munchies, and take something that can help with sinuses. if they start getting really fussy on the way up or on the way down, it's possible they have a sinus block. do what you can to clear the sinuses, as the pressure is blindingly painful (i've had it happen to me).
good luck, and if you hve any more questions, feel free to aks them and i'll check up on this thread.