It sounds like we have almost enough Brothers here to form a "Cooking Lodge of Research"! I know of at least 3 others who have not chimed in ... one in Australia, one in England and another in the USA. The question is under which Juristriction do we file for it??
THE APRON: Like Uncle Bob said ... the apron a man receives when he becomes a Master Mason (3rd Degree) is made of lamb skin. If you look under the flap it will have the name and location of the Lodge, his name, and dates for Initiated, Passed and Raised (the dates he passed the three degrees). Generally, this apron is not used in a members funeral/burial ... the lodge will generally supply a cloth apron for that so that his "presentation lambskin" apron can be retained/preserved by his family. I have the aprons for my G-Grandpa, Grandpa, Dad, and me. They make display cases (unfortunately my catalog is still packed somewhere or I could point you to a good site for them) - like they have flag display cases. Unfortunately - G-Grandpa's and Grandpa's aprons were not stored properly, plus they are old, and will have to undergo some "restoration" work before I can put them into a display case.
THE SHRINE RING: As far as I understand it (and I just talked to the Secretary of my Lodge who is a Shriner) - yes, you can wear it on a chain around your neck. If you know what Shrine Temple your Dad was a member of you can give them a call and ask for a definitive answer - or call a local Shrine Temple in your city/state and ask them.
SECRET SOCIETY?: Well, if you know it exists because it's existance is not hidden and is more than just myth or hear-say or legend - it's not much of a secret, is it? Walk around town and look at buildings like courthouses, libraries, schools - you'll find a lot of cornerstones with Masonic emblems and other information on them. Do Masons have "secret" ways to identify each other as being a "real" member? Yes. So did the Knights of Pythias, my fraternity (Alpha Phi Omega) - I was engaged to a Tri-Delt once and she never revealed their secrets to me - and I'm sure the Knights of Columbus have a secret or two.
DEGREES: The Masonic Lodge (aka the Blue Lodge) has only 3 degrees. After the Blue Lodge you can go up to 32nd degree in an appendant body ... one of the Rites - York (Commandry) or Scottish (Supreme Council). 33rd Degree is a very special honor ... it is not earned like the other degrees - it is a very special honor bestowed by peers upon a person who has given some "above and beyond" contribution to the community or society. After the Rites - you can go into the Shrine ... but that doesn't make you any more of a Mason than if you don't.