Does anyone have a theory as to how Usagi's dad could non-chalantly take the Silver Crystal to the jewelry store without it having set off alarms from Beryl and the like? But more importantly, how'd they manage to mount it considering that the Moon Sword was made of a material stronger than a diamond.

I think the mounting thing was just a writers flaw, and I guess the enemy originally had all shops on alert for the ginzuishou but by the time it was found their plans had changed to go after the princess and kill her and wherever she is the crystal will be instead of continuing to look at all jewelers.

I don't know about the second part, but for the first part, remember that part of the Silver Crystal's power was stored inside of Mamoru at the time so maybe it didn't have enough power to set off alarms.