Carry on for domestic AA and UA flights can be put in the cargo bin?

There are really 4 ways to get your bag onto the aircraft. Only one of these ways does not involve carrying your bag with you all the way to the boarding gate. That way is to check your bag at the airline check-in counter. Many airlines, including UA, AA, Delta, and U.S. Airways, charge for this service. For U.S. airlines that do charge, the most typical fees seem to be $25 for the first bag and $35 for the second. If either bag is over 50 lb., there will be an additional overweight baggage charge. The most notable exception to this is Southwest Airlines which does not charge anything for the first two checked bags, but does still have a fee if either bag is over 50 lb.

The other 3 ways to get your bag onto the aircraft all involve you carrying your bag up to the point that you are actually boarding the aircraft, including taking the bag through the security checkpoint. All of these methods require that the bag meet standard carry-on size specifications. They are as follows:

1. You can take the bag onto the plane with you as a carry-on item. Most airlines do not charge for this (Spirit and Allegiant do charge for this, though.)
2. On smaller 'regional jet' aircraft (such as those operated by carriers marketed as 'American Express', 'U.S. Airways Express', 'United Express', or 'Delta Connection',) larger bags that would fit in the overhead bins on a normal-sized airliner will not fit. As such, these bags will be collected on a cart at the gate as you board the aircraft and then returned to you as you leave that same plane. These bags are stored separately from regular checked luggage, but are not carried into the cabin with you. This service is not typically available on normal-sized aircraft, though, except for special types of items like wheelchairs, strollers, and such that people need to take all the way to the gate.
3. On many flights, there are more people wanting to take larger carry-on bags on board that there is space for those bags in the overhead bins (especially since airlines started charging for checked baggage.) If the gate agents anticipate this being the case for a flight, they will offer to check your carry-on bags for you at the gate, typically free of charge. This is different from what happens in case 2, as these items will not be returned to you as you exit the aircraft. They will be checked and placed with the rest of the checked luggage. You will not see them again until you get to the baggage carousel at your final destination. If not enough people volunteer to have their carry-ons checked at the gate, then the last few people who can't find space for their carry-ons on board will be required to check them.

The answer to both of your questions 2 and 3 is that unless you check them in (and pay fees accordingly,) they will fall into one of the 3 categories of bags listed above and you will have to carry them all the way to the gate, including any time that you may spend outside of the airport before the flight leaves.