When Castle hands Alexis her cocoa she has a pencil in her left hand but as they walk out the break room she doesn't have the pencil and she clearly doesn't put it down.
The killer used what one of the detectives describes as a "poor man's silencer" so that nobody heard the shot - sticking the muzzle of the gun into the neck of a 2-liter plastic soda bottle and firing through the bottom. This will, in fact work - for a 9MM or smaller automatic pistol. But the gun used turns out to be a heavy revolver - probably a .357, which is too powerful for the trick to work. A revolver cannot be effectively silenced, anyway, because the gap between cylinder and frame allows some of the blast to escape.
At around 13-14 mins, Castle remarks that he hid in the bushes on Alexis' first day of preschool. Capt. Montgomery remarks "first time my son went to summer camp, I followed the bus all the way to the Adirondacks!" Capt. Montgomery does not have a son, only 2 daughters, which is revealed in S.3 E.24.