I'm a claims adjuster, some good info in this thread and some myths.
First, as general knowledge; homeowner's policies are heavily regulated and 90% the same, nationwide no matter the company.
Contents have a lot of sub-limits, most of them are only for theft and apply to higher value things like firearms, jewelry, art, imported rugs etc. If you think about it, it makes sense, I can call up and say anything I want to got stolen, prove it didn't! Now, we will require you file a police report if you are claiming theft, but many people are willing to file a false report. We will also require proof of ownership, but will take anything, photos, receipts, call up the place of purchase, whatever. You don't have to establish the value, we will price it based on current value for a like quality item.
The firearm sub-limit is typically $2500 so the scheduled floater is to increase that sub-limit for theft. If you house burns down, we will pay the full amount of the firearms up to your policy limits.
The sub-limit for "electronic data processing equipment" (computers, smart phones, games systems and their software) is typically $5k total no matter the cause of loss.
Over a certain loss amount (varies by company, typically $2500 or $5k, maybe up to $10k), all policies will pay actual cash value (the depreciated amount) up front whether you have replacement coverage or not. On the items you actually repair or replace (with replacement cost coverage), send the replacement receipt and we send the rest of the money. For your old junk, it is your lucky day, keep the ACV payment and do whatever you want with it.
This is "indemnity" (making you whole), it was used stuff that was damaged not brand new stuff so you are getting the used value for the items. That is fair as you got years of use out of them already. Again, almost everyone has replacement cost coverage so you can replace everything and not be out of pocket if you want to.
This also applies to the home structure, ACV up front. I have had some people just pocket the cash, not rebuild the home and sell the lot (obviously paid off, no mortgage).
The sub-limit for cash/precious metals etc. is typically only $200 (for any cause of loss), so keeping a lot of cash on hand is at your own risk.
You could keep photos of your firearms collection, receipts and/or appraisals for high dollar items in a safe deposit box or fire safe. You could also take a couple photos of each room in the home, this will establish generally what you own.
I can't speak for all adjusters, but I only handle large losses (typically $100k+), I hardly ever ask for ownership proof, usually I can inspect the home and get a really good feel for what is there, even with a burn to the ground talking with customers I can get a feel for what they had. If it is reasonable I just price it out and pay for it. I need to see something for high dollar and out of the ordinary stuff, but customers usually have no trouble supporting they owned those items in some way.
Filing a property claim where your home/contents are damaged and can be inspected should be pretty straight-forward with little drama. Theft is where we see most of the fraud, so if you have a theft claim, file the police report and be prepared to give the adjuster some ownership support. Not for everything, but the bigger items. They want to believe you because not believing you is way more work for us, investigations, delays, mad customers etc. We want to get a reasonable claim, a reasonable list, and just pay it...