It may be more innocent. The lipstick name is probably taken from the first line of the Marine Hymn
"From the Hall of Montezuma, to the shores of Tripoli...."
https://en.wikipedia.org/wiki/Marines%27_Hymn
The red Scarf? It could hold some significance, or could just be a scarf.
That’s the red scarf picture I found! and good job on finding that picture of Reese with her lipstick!