What server infrastructure are they built on? especially with so many clients connected and communicating in real time.
I'd guess the servers will be running on Linux, BSD or Solaris almost 99% of the time.
Do they manage with scripts that execute on page requests? or installed services that run in the background and manage communication with connected clients?
The server your client talks to will be a server running a daemons or service that sits idle listening for connections. For instances (dungeons), usually a new process is launched for each group, which would mean there is a dispatcher service somewhere mananging this (analogous to a threadpool)
Do they use other protocols? because HTTP does not allow servers to push data to clients.
UDP is the protocol used. It's fast as it makes no guarantees the packet will be received. You don't care if a bit of latency causes the client to lose their world position.
How do the "engines" work, to centrally process hundreds of conflicting gameplay events?
Most MMOs have zones which limit this to a certain amount of people. For those that do have 100s of people in one area, there is usually high latency. The server is having to deal with 100s of spells being sent its way, which it must calculate damage amounts for each one. For the big five MMOs I imagine there are teams of 10-20 very intelligent, mathematically gifted developers working on this daily and there isn't a MMO out there that has got it right yet, most break after 100 players.
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Have a look for Wowemu (there's no official site and I don't want to link to a dodgy site). This is based on ApireCore which is an MMO simulator, or basically a reverse engineer of the WoW protocol. This is what the private WoW servers run off. From what I recall Wowemu is
However ApireCore is C++.
The backend for Wowemu is amazingly simple (I tried it in 2005 however) and probably a complete over simplification of the database schema. It does gives you a good idea of what's involved.