Does this sign mean the language or the nationality?
This sign is wonderfully versatile! You can use this exact same sign to refer to the Chinese language, a person from China, or Chinese culture and food. The context of your sentence will make your meaning clear.
Do I use my right or left hand to sign this?
Always use your dominant hand. If you are right-handed, start on the left side of your chest, move to the right, and go down. If you are left-handed, do the mirror image: start on the right, move left, and go down.
Is there an older sign for Chinese?
Yes, there was an older sign that referenced the shape of the eyes, but it is considered highly offensive and racist today. The sign tracing the traditional clothing across the chest is the only appropriate and accepted sign to use in modern ASL.