What I'm talking about is someone that generates paper wallets using something like bitaddress. Do they need to check every single address or just a few of them and then as long as those check out, they can assume bitaddress works correctly so there is no further need to keep checking newly generated addresses. I would think "yes" with a few caveats but in general "yes".
Up to the individual, probably. You need to differentiate between whether you are checking for accuracy or checking for maliciousness. If your copy of bitaddress generates one address accurately, then you can be pretty sure if it is non-malicious then it will generate all addresses accurately. However, a malicious piece of software may generate one or two addresses accurately and then start generating fake addresses.However, given that any malicious software could generate addresses from predetermined seed phrases or private keys which are known to an attacker, then testing them in this manner does not protect against attack. That can only be done by reviewing the source code.
What if the software I used uses a non-standard encryption and it turns out I can only decrypt it with that software? Then, 20 years later, I can't find it anymore.