What is the significance of the necklace Ash gave his girlfriend?

Yes. in a former version of their script, the glass was supposed to catch the sunlight and burn The Book of the Dead.

They probably dropped that idea since sunlight in the morning is very weak (perhaps the time when they wanted the glass to "shoot" the beam) and even at the apex of the sun's height, it would look a bit ridiculous for that little glass in Ash's hand to be burning a hole in the book.

"I must-I must experience the greatest act of a human mind:to take another life"

It's a macguffin that serves as a plot device for the finale.