The one challenge is that when you have "rare/collectible" items - like all your vintage stuff - that was originally $2 in the store and is now worth $125, your insurance won't generally cover that "collectible" value. I know you've got a ton of nice stuff that falls in to that category.
As I recall you would need to get your collection documented/appraised and generally get a rider on your insurance that would cover these collectible items. Depending on the coverage, it may or may not be all that much.
This is what I found for my insurance as well. They'd give you the $ you paid for a Star Wars figure originally, not whatever it is "worth". Like Dave said, for 90% of Modern era stuff, you'd be fine since you can replace a lot of things for less than what you paid. For the actual, real "valuable" vintage type stuff it's in your best interest to have a talk with your insurance agent.
I have an insurance policy rider on my home owner's policy for the "valuable" parts of my collection. It's not a big rider since I don't have a ton of "valuable" type items, so it's pretty inexpensive (I think it's $47 a year extra). It's basically enough that I could replace those truly valuable items I own at the full value it would cost to replace them.
I don't recall what all I had to do (I think it was just some pics proving I owned "collectibles"), I just remember it wasn't much. And the rider is for "Star Wars and Comic Collectibles" so the actual contents of the collection can vary a bit over time. I would need proof though if I ever had to use it, so I try to keep my collection spreadsheet current and I take pics of the "valuable" things I own in case I need them if something happens.