Can we still get a sense that we're playing a normal Civ game?
First off, what's "normal"? That's an incredibly difficult question to answer.
Secondly, a qualified "yes", up through and including deity but probably not Sid. At deity, there are still LOTS of ways to win -- military, tech lead, suck-up-all-game diplo win, etc. etc. etc. You can do things like a 5CC and actually be competitive and beyond in terms of research or pure cash or military, depending on your focus. Yes, the quality of play has to be higher (things like micromanagement, worker management, diplomacy, kill ratio, etc. become much more important), but I think it feels like any other game...you're just behind a lot and the AI stacks get HUGE. My experiences on Sid have led me to conclude that most of the time the way to win Sid is with a lot of military.... But that could be my bias. I'm certainly not aware of anyone getting (and holding) any kind of a tech lead on Sid, but that's to say it's impossible....
But, if "normal" means always ahead in techs and leading the opponents in every facet of the game...then the answer is no. For some people (no blame here, just saying), a game where they're not way ahead isn't "normal" or fun. If that's "normal", then, no, upper levels will never be normal. You start in a hole and it can get quite pronounced at times. Trying to catch up is the best part for me, though, because once I'm caught up, I've won, essentially.
My $.02,
Arathorn