Do slightly elevated liver enzymes always mean cancer? I'm only 20 and don't want to be worried.

Not likely to be can. Cancer is rarely the cause of elevated liver enzymes...So you do not have to worry that much. Yet they imply liver damage of some degree, the commonest cause is alcohol. So i would advise you to be sure not to drink alcohol till the enzymes return to normal. Also make sure that you do not have hepatitis. This can be done with a blood test for hepatitis b and hepatitis c.
No. Elevated liver enzymes can be caused by many things, meds, foods, infections, stones, etc. It's not very specific to any one thing. Many times they will resolve on their own. Just discuss them with your doctor.