Here are the combinations where mate can be forced: K+Q vs. K, K+R vs. K, K+2Bishops vs. K, amd K+B+N vs. K. It is possible in some cases to force mate in a K+N+N vs. King IF the opposing side has a pawn. These basic mates are explained in many good Endgame books including Reuben Fine's huge work "Basic Chess Endgames".

It is hard to predict what will happen in 100 years, but I think it would be safe to say that computers will have most of the openings and endgames worked out, which will wput more emphasis on memory in human competition. Maybe we will all be playing Fischer's Random Chess, who knows?? Or better yet, the Global Opening may be the standard way to get your opponent out of his book on move ONE!

I would say that there is no specific nuimber of moves one should look ahead. In a highly tactical situation you may have to consider a lot more possible "candidate" moves, and the length of variations you are considering may reach quite far. In a closed position you will be looking more at general considerations than specific variations. As far as the center goes, I would not say it is THE most important part of the game, but it is definitely one of the most important factors.