Should I get eyelid surgery with lateral tarsal strip or a skin graft?

Your images show a downtuning canthus, prominent eyes, a relatively flat cheekbone,  and maybe some mild lower eyelid excess horizontal length.  It is possible that you might have a congenital condition causing this configuration, with abnormal findings on an exam. It's also possible that your exam would be largely normal and that this configuration is simply your genetic heritage. 

The repositioning of tissues you are  demonstrating by pulling the corner of your eye towards the bone with your finger can be accomplished  with surgery, but the specifics of how this would be best accomplished would be  best determined at an in-person consultation with an eyelid specialist.  Options include non surgical fillers, a canthoplasty/lateral tarsal strip with recession of your lower eyelid retractors - possibly with a graft, or even a midface elevation. 
Again, it would be best to meet with a few surgeons to get a good look at your anatomy and options. Good luck!

Thank you for your question.  It does appear that the contour appear lower eyelids is such that you would benefit elevation of the lower lids, especially over the outer portion.  I would not recommend skin grafting for a number of reasons.  While an examination person will be required to make a definitive plan, elevation of the lower lids with a graft inserted inside the lid is the approach that is most likely to give you satisfactory improvement in your eyelid position.

While the skin graft approach is effective in helping to elevate the lower eyelid, it is highly disfiguring.  The skin graft simply never looks right.  For that reason we have develop the use of graft material inserted behind the lower eyelid to minimize evidence that such an extensive reconstruction has been performed.  The big problem that you prior surgeons did not deal with is the shape of the lateral orbital rim.  I approach this work with a hand carved ePTFE orbital rim implant and vertically lift the midface.  Additionally a hard palate graft is used to control the shape of the eyelid.  A lateral canthoplasty is used to close the area but it is not used as a structural element of reconstructing the eyelid.  The canthoplasty simply does not have the strength to do that.

froould not need a skin graftm your photos a canthopexy would change the position of your lateral canthus and bring it up. Unless you have skin loss under your eye from prior surgery you sh

This depends on several factors.  However, from just looking at the pictures, I would likely suggest a spacer and a lid tightening procedure.  The best advice I can give you is see a surgeon who specializes in these types of cases as they can be challenging to achieve excellent results.  Good Luck
Dr Z