Is Sleep broken/too powerful or simply another viable aspect of Pokemon? Is it overcentralizing?
I don't think it's broken. It's very easy to see it coming and pretty counterable. I run Safeguard on Cresselia which helps a lot, and I also find that sometimes I can leave the sleeping Pokemon in until it wakes up using Follow Me Togekiss as a partner. There aren't that many stat-boosting mons in Doubles so Togekiss just has to eat a couple attacks as long as they don't have multitarget moves. If the opponent goes for double sleep, it will likely take more than two turns unless they're using something Togekiss can seriously threaten with Air Slash/Heat Wave (Amoonguss, Breloom). As for Mega Venusaur, if they lead I can lead with Mega Kanga and Togekiss and use Fake Out and TWave to neuter it and then Air Slash flinch it until it has to switch. Also obviously Fake Out gives me a chance to set up Safeguard. But moving away from my own team's counters, there aren't that many mons that can use sleep effectively so it's predictable, and fast and powerful Pokemon are everywhere in Doubles so if you predict correctly you can get a KO without suffering through sleep.
Does it restrain teambuilding too much? How much preparation does a team require to combat sleep?
I think it puts a premium on Fake Out, Follow Me (specifically on Togekiss or Jirachi with a psychic move), and Safeguard, but these are very useful moves anyway. I originally put Safeguard on Cresselia to guard against burns. I have been fine without using a dedicated sleep counter that isn't effective in other areas - you can counter sleep without using something that would be bad in a Sleep Clause metagame.
Does Sleep limit or increase diversity of the metagame?
Darkrai is really only around because of sleep. One Poke that it makes a big difference to is Amoonguss, and I've even seen a few Parasects (although none that were used well - one was on a TR team with Surf Slowking which was kinda cool). Again, I don't get the sense that I cannot use certain otherwise-good mons because they're weak to sleep, or that I have to stick sleep counters on my team that aren't also useful in other situations.
Does Sleep have an overall positive/neutral/negative impact to the Doubles metagame? How does Sleep affect the metagame?
I'd say neutral to positive. It adds another strategy to toy with, another element to have to deal with. It doesn't necessitate a specialized, otherwise-useless counter. Seems good to me.
Is there any opportunity cost in using Sleep?
Yes, you rely more heavily on luck. It's hard to build a watertight, guaranteed-to-win team that prominently features sleep because things will just wake up after one turn sometimes, and one turn is all that's necessary to paralyze your sleep mon, KO it, etc. Almost all sleep mons are also grass type, which means you have to carry extra support against Talonflame, Charizard-Y, Heatran, etc. You also typically need a Ghost or two to deal with Fake Out, and a lot of sleep teams go for Trick Room which is limiting. I'd say Mega Venusaur is the main thing that doesn't require a lot of support to induce sleep, but it's also just very obvious.
What are the different checks or counter strategies to combat sleep?
Fake Out, Jirachi/Togekiss, Safeguard, Taunt, Protect, Pranster Taunt or Safeguard, Sap Sipper, grass types, fast and powerful mons in general, obviously Lum Berry and Safety Goggles, and just seeing it coming and playing smart. Doubles is full of really powerful mons so if you predict correctly you can get the KO before sleep comes. There are also just very few sleep Pokemon who also pack a powerful punch without setup, so leaving something in that's asleep can work out.
Conclusion:
Sleep clause should be lifted. It affects the metagame positively by introducing a new (or rather more viable) strategy element, the sleep inducer.