Is it legal to use handheld radios to listen to (general aviation) airport frequencies in the USA? Can I go to an airport with a handheld radio and listen to their frequencies?
Your receiver must comply with USC 47 302a, the law that says, "you can't interfere with other equipment".
USC 47 301 in general is all about "transmission of energy", or broadcasting. And it really is beyond the scope of government to say you can't stick a piece of metal in the air and process the signals it receives, provided you don't cause interference to anyone else.
Assuming your radio is receive-only, you really won't have a problem with that. (short of a total malfunction!).
Sure you can. I have an ICOM AC-24 Aviation VHF Transciever in my flight bag and will use it, often when walking around on the apron of untowered airports to be aware of, and communicate with, other aircraft on the ground and in the pattern using the CTAF.
Just be courteous with it and don't use the device to either interfere with ATC communications or for personal conversations with ATC or aircrews.
If you don't have the appropriate license and thus can't legally transmit, disable the transmitter. The easy way to do this is to open the radio and remove the little plastic piece that protrudes through the case on the transmit switch. Yes, you could push the switch with a pencil or other item, but disabling the transmit will show good faith that you are not breaking the law and have no intention of doing so in the future.