"One, I should be looking more at the mare's GSV and trying to raise her resulting foal's score, moreso than really looking at the stallion's GSV."
Yippee, thats right. Move up, move up, move up. With the caution that young, untested stallions who become successful later will move up that GSV score of the foal even years after it's birth.
For example, before Mr. Prospector produced a single foal to race had a GRA of 36.14 [note: Stud Performance adds the SP, which makes it the GRASP]. Add my normal default for a stallion of 6.36 in Stud Performance and mated to my mare A Screen Classic, their foal would yield a GSV score of 63.14. However, if I mated A Screen Classic to Mr. Prospector late in his career his GRASP score would be 78.54 and the GSV of the foal would be now 71.62.
One must remember, however, that most young stalllions fail so that many stallions only earn about 6 stud performance points. So it is possible that that GSV can go down if the stallion really fails.
"Two, I should be thinking more along the lines of the resulting foal's GSV and how that can be changed instead of concentrating on the GSV of the mare herself."
Yep, you can't do anything about her GSV unless you own her dad or mom. You can cull if the scores are low and her foals not what you had hoped. But otherwise, concentrate on how you can raise the GSV by using a stallion that looks like it will improve its GRASP scores in the future and also meets your criteria for type to type, conformation, etc. Never rely on the GSV itself. It is just a tool to help you easily detect bad matings in the long run and give you a heads up on good matings.
Hope this helps. The Green Beer of the Irish helped facilitate me this response.