How come Infinity Ward doesn't offer a hardcore playlist for every game type?. Recently they added hardcore domination, FINALLY. But my question is, do they have to have thousands of servers to support each game type or are they just restricting the playlist types for no reason?
They pick gametypes/playlists at their discretion. I'd suggest consulting the Call of Duty forums if you'd like to make a request. New playlists are announced on the forums as well, see this thread from December 16th about the addition of a couple of new hardcore playlists, for example.
Most multiplayer games of Modern Warfare 3, including all console games, are hosted peer-to-peer, where one player is acting as the host. In this scenario, servers hosted by Infinity Ward/Sledgehammer Games are only involved in matchmaking. Your system talks to these servers to determine which lobby to join and to find other players interested in playing the same game type who live in the same region as you do.
I have no facts about how many servers are running in order to support multiplayer matchmaking, but I do not believe there are thousands per gametype. An educated guess would be dozens or perhaps hundreds total across all gametypes. They likely have the ability to scale this number up and down as the game grows or wanes in popularity. I imagine that there is little load on the matchmaking systems per-playlist, and rather the load varies depending on how many people are actively searching for games at a given time. This is a hypothesis, however, and there's no way to know the facts without asking the developers directly.
On PC, it's possible to run a dedicated server, which means that one computer is responsible for hosting the game every time anyone plays. However, you will be playing an unranked game. This means you can't gain experience to the next level on a dedicated server. Instead, the server administrator chooses what gametype is played, as well as what unlocks are available. I am not aware if Infinity Ward hosts any of these servers. The intent seems to be to allow independent server admins to host game servers if they so choose.