Unfortunately most of the apparent cases of this are likely because under Wine/Linux some of the more advanced effects and visual options aren't available, so the display resolution or quality are actually not equivalent to what you'd get under Windows. There are some comments here about running on cut-down optimized versions of Linux and that's quite possible, but these are really edge cases. The relatively poor state of graphics driver support on Linux is the main problem but it depends on your hardware.

It's possible, and if I recall, it has been known to for some games (this was a couple of years ago when I was more interested in Wine and gaming, and even then not the most recent titles). Depends on your graphics drivers, etc. but there were some operations that were more efficient through the translation layer and OpenGL than DirectX directly (probably version 9 at that point).

I have no idea whether that's the case for modern DirectX and more recent games though.

In some rare cases like L4D IIRC. However it's highly unlikely for the simple reason that most of Linux GPU drivers are inferior to Window's.

Around 10 years ago I would sometimes see higher framerates in games like HL2 but I'm pretty sure it was b/c I was running Gentoo and had all the cruft removed and had custom compiled everything for my hardware.

I don't think this was ever the case, but sometimes WINE benchmarks come pretty close to Windows, especially with optimizations like CSMT.

Years ago I did a small experiment. I ran Armagetronad (OpenGL open source game) on Windows and Wine, on latter FPS was few percent higher. I really don't know why, but OpenGL probably helps (I guess that calls are directly translated to native OpenGL).

Not sure about games, but I remember trying some benchmarks on Windows and under Wine (same machine) and it was faster on Linux

(but this was 10yrs ago)