Is It Common for a Lower Eyelid Skin Pinch to Alter the Shape of One's Eye?

Lower lid skin pinch can affect the lower lid contour and position if done aggressively and incorrectly.  It is not supposed to if done correctly, as proven in a paper I published entitled "Lower transconjuctival blepharoplasty with and without skin pinch" (referrence is on my website).  Consult an oculplastic surgeon.

Generally yes, removing skin from the lower eyelid has the risk of altering the shape of the lower eyelid.  For every action there is an equal and opposite reaction.  This can be even more dramatic because some doctors don't just do a skin pinch but use this term to lull individuals into having what is actually a much more radical lower eyelid transcutaneous lower eyelid procedure.  So generally you really need to know your surgeon before agreeing to a "skin pinch."  Consider reposting your question with photos for a more specific answer.

It can pull down one’s eye and change the shape.  Fraxel CO2 laser can sometimes help, but it can also pull the lid down more.  I would see the doctor and possibly get another opinion to see if any further correction should be done.

When done for the appropriate reasons (e.g., excess skin of the lower lid, as well as good lower lid "tightness," on exam by your facial plastic surgeon, among other things), the skin pinch does not commonly cause rounding of the lower eyelid. That said, over-aggressive skin removal can lead to problems, including change in the shape of your eyelids. 

A skin pinch blepharoplasty really should not change the shape of your eye.  If a canthopexy is performed to maintain support, then the eye may appear a bit almond shaped. This usually softens with time.

It sounds like you've exhausted through a number of conservative measures. Sometimes by performing Botox laterally around the corners of the eye to help diminish those fine wrinkles (rhytids), you then leave the muscle beneath the eye to become hyperactive and can produce greater skin folds and patient dissatisfaction. A skin pinch blepharoplasty or eyelid rejuvenation technique is typically performed in patient's who demonstrated mild skin fold excess without signficant lower eyelid festoons. Because there is limited skin removed, it should not alter the shape of your eye. My preferred technique is a transconjunctival fat removal (inside the eyelid), followed by a skin pinch anteriorly. However if you demonstrate appreciable skin excess, a more traditional blepharoplasty would probably work better for you. 

Unfortunately its not possible to assess your lower eye needs without an exam and certainly not without pictures. If you have significant skin excess or laxity, as well as fat herniation and/or festooning, you will need a traditional blepharoplasty with treatment of fat, a suspension procedure and skin excision. Any type of lower eyelid surgical procedure has a potential of changing the shape of your eyes and causing some rounding.