Does the hCG hormone produce quicker/slower depending on age? For example: a teenage girl in early pregnancy, one who hasn't tested positive yet.

Embryo. It is the embryo that makes HCG and it is not likely that the age or the women will make a difference. It takes about two weeks for the embryo to make enough HCG to produce a positive pregnancy test. Blood test may be done a day after a missed period. Home pregnancy tests should be done 4-5 days after the missed period. Please see your doctor if you are planning to be pregnant.
Pregnancy. Age does not affect the outcome of the HCG level .If you have a normal pregnancy it will double approximately every 48 hours. Positive test is around 10-15 days after conception ( just as an estimate).