Are they very serious and stick closely to actual history, like a NES, or are they more wacky and far-out, like War at Civfanatics?
Depends on the particular edition, though it's typically somewhere between NES/lazyNES and War@CFC, leaning towards the latter.
Also, how much behind-the-scenes diplomacy is there
A fair bit.
and how much of a time commitment is it to play?
Again, depends on the game, how "committed" you are, and the people who are playing. At one point, it took half an hour to just read the posts in IOTIV from the previous day. Currently, Mini-IOT would take up less than five minutes a day all up, IOTII and IOTV around ten. Take out the diplomacy, and it only takes around a minute a day to send in your claims or orders.