I use dx11 with dolhpin without issues. Make sure you use ubershaders and try with these drivers.
Okay, so trying ubershaders (asynchronous) without pre-compiling, I was able to get into the MKWii menu, but it becomes incredibly slow. With pre-compiling, it crashes like before.
That's before downgrading the graphics drivers to the February 2018 release, which I'm downloading now.
Edit: So the Feb 2018 drivers do now allow Metroid Prime and Mario Kart Wii to be playable when using asynchronous ubershaders, but the performance is also quite poor compared to when playing with the latest drivers.
Further, when playing Captain Rainbow, the save menu is very slow on the old drivers, but fullspeed on the latest drivers.
I'm going to put this together in a bug report for Dolphin's issue tracker soon, as there might be some issue with Dolphin's D3D11 implementation when using the current Intel HD graphics drivers.
Are you aware of any other problem titles aside from Mario Kart Wii and Metroid Prime that I should check? I have a fairly sizeable collection.
I also have a ThinkPad X220 I can test with since that runs a core i7-2640m, which features Intel HD 3000 graphics.