I just took a trip to India and got a sim card. The kiosks WERE open at 1am when we arrived (as was Costa Coffee) in Delhi (don't know about other airports). The kiosks are past immigration and baggage claim. They are in the last area before you actually go outside. We did need all the docs you mentioned and a local contact who could vouch for us. The phone company contacted him and made sure he was really authorizing us. The sim doesn't activate until they get that approval. Ours didn't work until maybe 3 or 4pm the next day (so that's shorter than the 3-4 days other people are saying). I needed to restart my phone to get it to work, while my friend's iphone just worked at some point.
We wanted data too, and that was the more difficult thing. It was actually very cheap compared to the U.S. (but also painfully slow) so I'd totally get it again now that I know how to the system works. The guy at the kiosk was not at all helpful and it was like we were asking something no one has ever asked (to get both phone and data for our unlocked phones). It should have been something he handles every day. Anyway, here's how it worked for us (not sure if this is how it is supposed to work or how it happened to work for us). We got a sim card at 1am Saturday night. He gave us a sheet with our Indian number on it and the phone number we were supposed to call in about 24 hours to activate it (which we did around 3pm the next day). Calling that number and getting it activated wasn't too much trouble, although you might need to mess with your phone's settings to see if it is helping you dial internationally, which messes up the area code (mine was doing that). The sim card came with some value on it (which the kiosk guy didn't tell us). We had plenty of minutes leftover, so definitely don't buy any more at the beginning (well, maybe clarify how much value you sim card comes with first). You can recharge (i.e. add minutes) at any snack store in town.
For the data, the kiosk guy said to go to any store and get it activated for data. The problem was, this wasn't true, and it was Sunday the next day. We couldn't just recharge it the way you recharge talk time. We had to go to an Airtel office and give our phone to the guy behind the desk, who messed with things and entered things and set it up. The Airtel office in our area was not open on Sunday and on Monday didn't open up until 10am (and then not promptly, resulting in a mass of waiting customers with no organization), so that was annoying. I wonder if it was all necessary or if the guy in the airport kiosk could have done it. The hard part was trying to figure out how to actually get things to work. You have no one to ask when your phone doesn't work because you have a phone that doesn't actually work.
In the end, it is inexpensive to get talk and data, and I would definitely recommend it. But, just know that it might require a little work to get things up and running.