I've had all those PC game problems also... but I've also seen the same games work as well without problems. PC is a different animal and it isn't a baby's toy like a console is where you just put the game in and away you go. Here are some examples based on my experiences.
- Games not boot: One example I can think is that some installers will let you install on incompatible systems, but then when you try to launch the game, it will crash. Cities Skylines has 64bit binaries, but the installer is 32bit and will allow the game to be installed onto a computer with a 32bit OS installed. This is because the installer incorrectly uses CPU architecture detection to determine requirements instead of the OS. Then of course, if you try to run Cities Skylines afterwards, it will not launch.
- Crash on startup: I've had this happen for sure, but I do not think I can clearly recall a game that did this all of the time. I have definately seen it happen a lot in modded games, but that would be the reason right there.
- crash randomly: Sure this happens for a lot of different reasons. The most notorious for me was Oblivion, but I've also had Pinball Arcade crash as well. One interesting game I remember not being able to play due to crashes, initially was Heroes of Might and Magic V. The crash would occur on a Radeon 9800XT but not on a GeForce N220GT.
- messed up graphics: Again, changing a video card fixes this.
- blank screen: Typically this happens when trying to run a game on a display that parts of the game do not support. There are various games where the UI and the game run on different engines, and the UI either will only run on specific resolutions or will only run on 4:3 displays, even though the game engine will run on any. This was a problem in the NHL series, starting with NHL 08 (and going back to at least NHL 06), which originally did not support widescreen monitors in the UI. So when you launched the game you would have no video. The "FE" had to be edited to allow the game to launch with video. This problem also exists for older games that used full-screen shells... most games do this now, but there was a time when it was definately separate. Those were games that did not support windowed mode. Sometimes these would not show video for similar reasons (resolutions or color depth incompatibilities).
- games running badly: all to do with system optimization. Some games will allow you to run them below specs because they do not check or check the wrong things. Notable Examples I can remember are Unreal Tournament running on Windows 2000 with 16MB RAM and no video card, getting 10FPS in the tutorial level. Obviously this game will run fine on appropriate hardware. Another example is Watch Dogs, which will run on a PC with a dual-core i3 and 16GB RAM, but it runs very badly. Change to a quad core and problem solved.