Have a look at the engine DB on devmaster.net
Also, all of the big engines nowadays are pretty much multipurpose... yes, you can make a strategy title in Cryengine. You might not be able to use all the FPS-oriented helper classes or tutorials, you might need to work around some other limitations, but its a good start.
Generally I would recommend starting out with Unity or Unreal Engine 4 for 3D Engines. They are the big commercial engines out there in the Indie-space, both have free or low-cost licenses available, both have huge communities and a large set of available tutorials and APIs, both are pretty much up-to-date (UE4 certainly is cutting edge, Unity will be again with the Version 5 coming out soon) .
About using different engines for each project: AFAIK most studios nowadays stay with their chosen engine for as long as it makes sense. Relearning the engine API and Editor takes time, you might need to redevelop needed plugins and editor tools you already have for the old engine, you might need to rebuy licenses for middleware solutions, and so on.
As said, most engines nowadays are pretty much multipurpose, and there is almost no engine out there that really allows you to develop withour writing all the game logic on your own (there might be tutorials, template projects or helper classes that provide you with part of the functionality, but none of this really is part of the core engine), so there is little functionality that one engine can boast another does not provide (For example, Esenthel Engine has built in Parallax Terrain shaders and Terrain Streaming, ideal for large openworld Games. You can do the same with thirdparty assest you can get for unity, its just not built into the core of the engine).
As an example:
WoW was built with the engine from Warcraft 3... of course, heavely modified (and back then there were no cheap options for engine licensing, so investing lots of money into making an aging RTS Engine work as Third Person cam MMORPG engine made still sense).
Today, this will be pretty common (All the World-of-... Games from Wargaming will be using the same aging Bigworld Engine WG licensed for World of Tanks. WG even bought Bigworld later on, and re-engineered the whole engine when most people were pretty much sure the engine itself was old and dated, and the only good part really was the netcode. They did not not dump the engine, even though there might be better engines out there nowadays for little money, they choose the old patched up Bigworld engine even for their newest game World of Warships. Shows how much of an effort they expect porting to a new engine would be).