Yep, like on an arcade :). Or, you could also leave the Genesis plugged-in and take a break (or let other people watch TV) and then continue on. As long as the console was powered, the game state was preserved. But this prevented you from changing games, or other people form using the console, so it was seldom done.

Surprisingly few Genesis games had any kind of save game feature - off the top of my head, there was Sonic 3, RPGs (which were few and far between), and...that's about it?

Playing the 32X port of Doom with no save ability was a pain. The game inevitably crashed on level 16, but it took my brothers and I several entire Saturdays getting that far to realize it. I loved the game, it wasn't until years later that I realized how atrocious a port it was.

Basically, yeah. A few sports games did as well, like some of the NHL Hockey games starting with NHL '95 I believe. (There was a "play a whole season" mode...)

Some classic games would generate a "code" you could enter to resume the game, all codes would deterministically map to some game state.

Ghostbusters for Sega Master System did this. One day I had a typo in my code and ended up with obscene millions of dollars.

I always wanted to go back and RE that game's code generation algorithm...

IIRC it was the same with many NES games, e.g. Super Mario Bros.

It was the same for arcade-oriented NES or SNES games, but not all games are such, both NES and SNES had numerous RPGs, or games with progress (SMB3, DKC) or unlockables (Mario Kart IIRC)

Yes Sir :)