First multiplayer FPS I played was Jedi Knight: Dark Forces II (released in '97). In that game, you had to lead your shots to a silly degree to actually hit anyone. But I think you're right; by then most games weren't suffering from that problem as much.

Yes and no.

Different games (really engines) had different models for it. Some games you would feel things grind to a halt while you waited for a packet. Others you would have rubber banding where the prediction of what your opponent would do was wrong and they teleport 2 meters to the right. And a select few would result in endless double kills as you both killed the predictions.

The big difference was that arena shooters (which DOOM effectively was) tended to have encounters where you might have 3 or 4 players all shooting each other at once with a high enough TTK that it was very easy to lose track of one enemy because you saw a more immediate threat. So it was a lot easier to just assume the rubber banding was a you problem or not notice it at all.

Then we had CoD and it all became about super short TTK and 1on1 fights. And now? Now it was incredibly obvious when someone warped because they were your only concern.

Back in the day, my games were UT (mostly the good one, sometimes 2k4), Jedi Knight 2, Tribes 2, and Operation Flashpoint. I was a cool kid... But even then, it was almost never perceptible in UT even though the Unreal Engine had "the worst netcode". Also not OFP since your encounter ranges were so long and you were squinting through iron sights so you had no idea if you missed because of lag or what. But JK2 and Tribes 2 were VERY obvious when the network was acting up because you were generally dueling someone or taking out a lone flag carrier while skiing across a field.