Well, I suppose the easiest way to go about a first post here is just to address the questions in the OP directly.
Is Sleep broken/too powerful or simply another viable aspect of Pokemon? Is it overcentralizing?
So, first off, do I think Sleep is too powerful or over-centralizing? No, definitely not. In my experience battling without Sleep Clause, it never really seemed to matter whether one teem tried to abuse sleep, both teams did, or neither. Regardless of the matchup, Sleep was very rarely a dominating force in a battle. That's not to say it was never important, or that the lack of a Sleep Clause never cause a game to be won by someone who would not have won it had Sleep Clause been into effect. However, the influence that Sleep has gained by enabling players to inflict it on more than one Pokemon is not all that high. Certainly, when compared with the other forces of the metagame, the effect Sleep has is nearly negligible. I think the most important thing to keep in mind here with regard to the power of Sleep is that Sleep with out Sleep Clause does not need to have no real increase in power for it to be worth allowing. Sleep is definitely stronger than it ever was before. However, I do not think that the power it provides is anything unacceptable for a metagame like this.
Does it restrain teambuilding too much? How much preparation does a team require to combat sleep?
I personally never really found that the threat of Sleep caused me to change anything I did when making teams. Now, to be fair, I am definitely more of a very quick build, try, and adjust player than one who spends hours thinking about what to put on a team. Yet even so, never did I ever go back to adjust a team with the intent of making it more prepared for sleep. In almost every battle where I face off against someone trying to use the lack of Sleep Clause to their advantage, it was the way I played against them that had a much bigger impact on how successful their sleep strategy was than any individual part of my team itself. The common natural immunities, such as the grass type immunity to spore moves, and common weaknesses of sleep inducers, play a big part in this, as I am more likely than not already going to have something to use against Sleep on any given team. Is it theoretically possible to to have a team that is overly weak to Sleep abuse? Sure. However, I don't think it is any more likely than a team that is naturally weak to one of the hundreds of other normal elements of the doubles game.
Does Sleep limit or increase diversity of the metagame?
I think diversity has only increased by the elimination of Sleep Clause. An increase in use of more dedicated sleep inducers, such as Jumpluff, are the most obvious increases in diversity, but I do believe that it really does go far beyond that. Overcoat, which was not an incredibly useful ability, now can be very handy, increasing use of things like Escavalier. Grass types in general, whether or not they carry Sleep moves themselves, got a boost, due to them being immune to something that has become more common. These are all fairly minor increases in diversity, but they are definitely real. What's more, its not like anything has become less viable to counterbalance this. As others have mentioned, for the most part, you could very easily use almost any team that was made for the Sleep Clause meta and have it work out nearly as well without the Sleep Clause. Nothing has become less useful, other than via acceptable levels of normal metagame drift, while multiple Pokemon have become more viable. And I'd even argue that the diversity increase goes beyond just the Pokemon themselves. Items have seen a great increase in diversity. While, as I mentioned, no team necessarily seemed to have any severe problems with Sleep, one could very well take advantage of Safety Goggles or a Lum berry to turn a situation very much against someone trying to abuse sleep. This is not a case of being forced to run a counter, and I vehemently disagree with people who claim otherwise. And no Pokemon shouldn't be running these items. It's simply a case of people being given an option to try and use their item slot in a new way to potentially give them advantages that were previously unavailable.
Does Sleep have an overall positive/neutral/negative impact to the Doubles metagame? How does Sleep affect the metagame?
So overall, what is the effect of the removal of the Sleep Clause on the metagame? I think it is a net positive. It is not that big of an effect, but it is there, and I think it is a good thing. It increases diversity, without really having any real negative effect. I don't have much more to say on this one. Everything was pretty much covered by the last 3 paragraphs.
Is there any opportunity cost in using Sleep? (EDIT: Nice wording change on this one)
Well, pretty much by definition, unless you are choice locked / encored into a move, cannot switch out, and can only choose one target, there will be an opportunity cost. Further, in the game of Pokemon, it is very rare for the maximum possible opportunity cost not to exceed the most probable outcome of choosing a given action. For example, say you are using a Venusaur, and you think you will need 3 attacks (2 from partner and 1 from Venu) to take out an opponent, and you don't want the opponent to be able to do anything before that happens. Using Sleep Powder might be, on average, the best way to take it out, but by going that route, you forgo the chance of getting double crits and a Poison from Sludge Bomb, which might do the job in a single turn. In other words, yes, Sleep has an opportunity cost, but so does everything else. The real question is whether or not, on average, using Sleep moves has an average opportunity cost greater than or less than the returns provided by the sleep move itself.
This is something that I think is really hard to measure, but I think I can say with some certainty that the average returns from sleep moves are definitely not so much greater than their average opportunity cost that their existence is a bad thing for the metagame. To put this into less economic terms, I think sleep is not obviously better or worse than any other given strategy on a given turn in your average battle. Beyond that I think it is really hard to judge. There is however one thing I think I can say regarding this: I believe that the usage of sleep moves has diminishing returns to scale. The advantage gained by Sleeping one opponent is significantly larger, on average, than the advantage gained by Sleeping a second opponent. Passing up a 2 on 1 scenario to vie for the possibility of having a 2 on 0 scenario is frequently not worth it. Sometimes it is, but I don't think that could be called the normal situation.
What are the different checks or counter strategies to combat sleep?
Honestly, I don't think there are much worth mentioning, because you really shouldn't need a specific strategy or check in order to beat sleep. I pretty much agree with the sentiment of Pwnemon's recent post in the other thread that, really, the best counter to sleep is being a smart player. Smart switches, subs, protects and the like are by far the easiest way to beat any player trying to take advantage of Sleep. I personally feel that if you are running anything on your team specifically to counter sleep, then either you are not playing smartly with your team, or your team was just not up to snuff to begin with.
Are there any suspects independent of Sleep that you want to raise attention to the council?
Well, I kinda want to say something to do with sun, but I can't really pinpoint what. I don't think sun itself, or even Zardy or anything is specifically too powerful, but something about that overall strategy makes me feel like it might be worth a look at. However, as far as individual suspects, I really got nothing.
