Can I travel between different states in the USA by air (or land) if I lost my Hong Kong passport?

If you lose your passport or it's stolen in LA, your best bet would be to file a police report (to document the loss/theft and to have official proof of the situation, which is very important) and contact the Chinese Consulate in LA for assistance. They'll be the ones to get you a replacement passport. You can also call the Hong Kong Immigration Department's 24 hour hotline (scroll to the bottom) for advice and assistance.

This is actually much more convenient than if you were in South Carolina, as there is a Chinese Consulate in LA, but none in South Carolina.

Since your passport contains your US visa, you'll also want to report the loss/theft of your US visa to the State Department. You'll be able to remain in the US, but once you leave the country, you'll need to receive a new visa at a US Embassy or Consulate before you can return, just like you got one originally (you cannot get a new student visa from inside the US). You'll want to talk to the international studies office at your school as well, since you'll need paperwork from them for a new visa. You'll want to plan any future trips abroad around this, since you'll need an appointment and sufficient time to get a new visa, and you don't want to be stuck outside the US.

You can travel by land without your passport, though the ID requirements of any bus company or rental car company still apply. To travel by air, you need to show a TSA-approved form of ID. If you have a US driver's license or state ID (not a student ID), that's acceptable. Without an approved form of ID, you may be able to travel by air if you bring all the documents you can find (including the police report mentioned above) and undergo extra security screening, but this is at the discretion of the TSA; there's no guarantee you'll be allowed to fly under these circumstances.

If this happens, replacing your passport in LA is probably the best option, since you'll need to travel to a consulate for the replacement anyway, and there's already one there.

Have a great trip!