As with all things Civ, the answer usually is "it depends".
Reloading is certainly cheating in a competition game like the GOTM. I've never done a succesion game, but I would say it's very frowned upon in them as well, unless all the team members agree to.
If you are playing to better your skill its usually ok if the reload helps you learn from your mistakes. Like when to declare war, or how many axes to bring in a rush, or where to settle to beat the AI to a good spot. In a public game, like the student series, reloading I think is ok, but you should probably be honest and tell everyone where and why you reloaded.
Finally, for a single player game where you are just playing to have fun, my personal opinion is "anything goes". I only reload rarely because I feel better about myself when I can overcome challenging situation rather than try to make them not happen. But that's just the way I play, I won't judge anyone here for playing the game their way if they are having fun.
I'll admit just last night I reloaded a game because I sent an axe to take out a barb city with 2 archers. He killed the 1st one and while he was healing up a German horseman came and took the city instead, which I was planning on razing because it didn't fit my neat dot map. I didn't want to declare war on Germany because we were friendly, so I reloaded the game and waited until I had 2 swords ready before burning the barb city.