Thanks so much for the help and support over the past few weeks/months. About two weeks ago my son and I were able to install our new liner, and everything went SWIMMINGLY.
No leaks! We made a new gasket for the opening to the skimmer hole, which seems to be holding up just fine.
All we have left to do is purchase and install a new light fixture (we have the bracket for it screwed into the wall and just need to get the fixture).
I hope I don't have to do this again for quite a long time, but really, this was my first time doing this and it went much, much, much more easily than expected.
Thanks again, folks! Oh, here's a photo I just took: