How can I travel outside the U.S. as a green card holder?

If you have properly filed the Form I-131, you don’t need to worry about how fingerprints are captured. The USCIS decides whether it requires new biometrics or should it utilize the previous ones; an applicant does not have control over this. Green card holders can travel in and out of the U.S. freely; if you stay out of the U.S. for more than 179 days, you will be subject to an admissibility inquiry. You need to have a valid re-entry permit if you stay out of the U.S. for 365 days or more.

You only need the green card to travel outside the U.S. The re-entry permit is only necessary if you plan to be out for an extended period of time, usually over six months.

There should no problem in leaving and entering the U.S. because the USCIS takes fingerprints.

You can travel with a valid passport and green card only. If you stay outside the U.S. for more than one year, then a re-entry permit would be needed.

If the USCIS notified you that an ASC biometrics appointment was not needed because they were able to reuse your biometrics, then it is fine. As long as you have a valid re-entry permit, you can remain outside of the United States. Without a valid re-entry permit, a green-card holder should maintain the intention to permanently reside in the United States and should not be absent for more than 180 days on any single trip abroad.