There were a couple of sites where this wasn't the case, or where the hand history wasn't provided in a simple text format (or at all) -- but generally, that's how it worked for the majority of the poker sites/clients.
The reason for being able to view mucked hands (at showdown) is because you can do the same in live games, at least that's the house rule in the vast majority of live rooms. It's almost always frowned upon, to varying degrees, to actually request to see a mucked hand - but generally IS allowed by rule (ie WSOP events explicitly allow asking to see hands mucked at showdown, though an 'anti-abuse' discretionary clause is included in that rule, too).
You will definitely get yelled at if you attempt to flip someone's cards yourself, 100% of the time. At higher limits, it may get you removed from the game or worse. You can ask/request the dealer to show the mucked hand, though. I'd still suggest that you not request to do so -- it will almost assuredly piss someone (or multiple people) off. There are some rare occasions where it may be worth it and/or warranted, but they are few and far between.
Most common is when someone shows their hand to "half" the table immediately before mucking (at showdown), which often will cause the other half of the table to want to see too. Most dealers handle it well and just move on to the next hand while the players 'disseminate' the (usually useless) information.
Do you have any reference for the rule that a folded hand has to be shown on demand if play is over?
See here for the Texas Holdem rules that WSOP uses: https://www.wsop.com/poker-games/texas-holdem/rules/
Search for "muck" to find the relevant entry, also below for convenience:
Any player who has been a legal part of the game can ask to see a called hand, even if the cards have already been mucked. Abuse of this privilege can result in denial by the dealer. When a winning player asks to see a mucked hand, that hand will be considered live, and the winning player can lose their pot. When any other players ask to see a folded hand, the hand will remain dead.
Only from the tables you're playing at. To my knowledge most sites won't allow you to spectate other tables for the purpose of gathering data.