Not a popular game, but the cartridge for Morita Shogi 64 had a built in RJ-11 jack and allowed online multiplayer.
Plenty of SNES games included a SuperFX graphics coprocessor in the cartridge. Star Fox is probably the most popular example. SuperFX games were popular enough that it was quickly emulated successfully. But you can't treat the cartridge as just a file system abstraction on SNES.
The instruction timing and such fiddly low-level details is most visible in systems programmed in low level, such as older consoles.
Modern systems tend to be programmed in higher level languages, which works much better to HLE (high-level emulate), since they tend to be programmed in a way that cares less/not at all about underlying hardware fiddly details.
So when you are emulating the NES, these details matter way more than when you are emulating the Switch for example.
Not really (and you don't always want perfect accuracy anyway).
You don't only need to validate the operations of the machine, but also the timing with which it operates.
And I don't mean just the clock/frequency/operations-per-second - it's everything.
Every little chip that performs a calculation, every little wire that sends a message.