Yes, you can gate-check the seats, I've done it many times. Tell TSA that they will be using them on the plane if they ask--it doesn't matter if they won't be; TSA does not police the use of carseats, but they sometimes complain about hand-wanding them if they are too big to fit through the x-ray machine. (If they do fit through, put them through, of course.)
Tell the gate agent that you wish to gate-check the seats because you didn't want to let them go through the bag room or risk the possibility that they would not get on the same plane that you do; they will be OK with that; things that go through the bag room get banged around a LOT more than things that are gate-checked.
When you check your other luggage, ask the SkyCap or Ticketing clerk for plastic bags for the seats; they don't keep the bags at the gates, and the hold of an aircraft can be a greasy place. They will give you large clear plastic bags that you can just hang on to until you get to the gate. Pop a little hole in the bag so that you can put your gate check tag onto a harness strap but still have it hanging outside the bag so that the ramp worker can read it clearly.
As to luggage allowance, it kind of depends, and you should ask the airline directly how it will be counted. In any case, trust me, if you are travelling with kids that young, you do NOT want to use up your entire luggage allowance anyway; you would have WAY too much stuff to keep track of.