It does NOT work or work well with DirectInput (older, legacy, pre-2006) games, only works with XInput games.
Before 2006, all PC games used DirectInput API standard for game controllers. Then in 2005, Microsoft implemented the XInput API for Xbox 360, and Windows had to follow and adopted the new Xbox 360 standard. However, the transition process for Windows has been long and gradual: some Windows games from 2013 still used DirectInput, (e.g., BioShock Infinite.) The Xbox 360 controllers do NOT have built-in functionality to work with games that use DirectInput. Sometimes the controllers may work but poorly: the controls are erratic, only partially functional, and/or either too sensitive or insensitive.
I found out about the DirectInput vs. XInput problem only after I had tried to use an Xbox 360 Controller to play BioWare's Jade Empire: Special Edition, which was released in 2007 on the PC. Jade Empire used DirectInput. The controller worked only partially: the left joystick (that controls movement) was moving by itself so my game character could not stand still; the right joystick (that controls camera view) and D-pad (for abilities) were non-functional. I went online to look for solutions, and learned about the different DirectInput and XInput API standards. That was when I found a long list of PC games, mostly older titles, that do NOT work or work well with the Xbox 360 controller. The solutions I found were either very technical and difficult to implement, and/or was an additional cost, (e.g., buying a gamepad emulation or profiler app, which usually costs around $10; none of the free ones work well -- I tried;) and/or was inconvenient to use, (e.g., every time I reloaded Jade Empire, I had to unplug and replug in the controller connection for a gamepad profiler to continue to work.) They are just hassles.
The problem with Microsoft Xbox 360 Controller for Windows, which none of the other reviewers/fanboys has even bothered to mention, is that it does NOT have a built-in function to work with DirectInput. It may work, but it certainly does not work well or perfectly. The Logitech F310, for example, has a physical switch behind between "D" (DirectInput) and "X" (XInput) methods. The switch should stay on "X" most of the time. However, if you ever run into a DirectInput game, you just switch over to "D". The Microsoft Xbox 360 Controller does NOT have that switch. Microsoft wants developers and users to conform: everyone should stop using DirectInput and adopt XInput, so Microsoft would tell you to "just don't play that game" instead of giving users a real solution. (That's that answer other people got when they called Microsoft to try to get a solution for the DirectInput problem.)
If you know you will definitely NEVER touch an old, pre-2006 game that works better with a gamepad or run into one of the odd new games that still uses the old API standard, then this controller may work for you. However, if you do run into a game that uses DirectInput, it will be a lot of frustration and hassle.
No, the PC version of Minecraft does NOT support controllers like this, however there are mods available that add support. Do a Google search with the following terms to quickly find info "Xbox Controller PC site:minecraftforum.net" That is the official Minecraft forum, so it is trustworthy and heavily patrolled for malicious content. Be sure to read some of the comments before downloading something, though, just to be safe.
I do not believe so, unless the game has controller support, the analog sticks do not take the place or act as a mouse in any situation. This could vary widely on the game, but in most cases this answer will be no.