Normally it should just mirror the main display, so maybe the busted display is more like a busted graphics card...

The LCD is "busted" as in physically, it has a large crack and impact damage on it.

I don't know about that model, but I remember that an older G3 iMac (the goofy colored ones with a CRT in them) was picky about what keyboard I used. It ignored my known working USB Dell keyboard and wouldn't work until I plugged in the original iMac keyboard. That was sad because the G3 iMac keyboard has such tiny keys that I consider it unusable.

I suppose this issue might not be universal - at the time I think that Dell keyboard was the only other USB keyboard I owned, so I don't think I tried any others.

Keyboard aside, I have no idea how to switch outputs, I really don't know how to use these machines at all.

I've definitely run iMac G3s on non-Apple USB keyboards and mice. Generally they work fine, just some keys have different functions. I recently picked up an iMac G4 800 for free. It booted exactly onece, then failed to boot. Dead PRAM battery (Apple genius: avoid having a 5V standy power line by using the PRAM battery for soft power on 😵 ), have ordered one. Once it arrives I can see what boot behaviour is with an external monitor. Nice coincidence: this afternoon I can pick up a free Apple Cinema Display, so I can even test Apple vs non-Apple screens. But no sign of that battery yet, so no answer yet either.

Edit: PRAM arrived today and so did the - huge - Cinema Display. Unfortunately I hadn't checked the rear of the iMac too closely - it has a mini-VGA and I don't have an adapter. At least, not for that. The Cinema display did come with a ADC-DVI adapter though (which was ironic as the donator said I might need some kind of adapter to get it to work - not realizing she was giving me exactly that). Maybe we still have a relevant adapter at work gathering dust somewhere in the cupboard, but I doubt it. They're cheap enough though so might just order one, but might be a while before I can say what behaviour should be...