Is it possible, by doing stretches that target and length the spine, to permanently gain about an inch of height?

Quick answer: No. You can improve your posture and thus measure as taller. The intervertebral discs loose water temporarily so we can measure shorter at the end of the day than in the morning. Also, with age we loose disc height, so old people get shorter. So, you can do things that will temporarily make you taller by maybe an inch. However, these are not permanent.
No . You can maximize your height by elongating muscles and strengthening your core to improve posture.