I agree with you that your backsplash looks very nice ungrouted. A color that looks like the ungrouted space!

Stay with a light to mid tone grout or you will start to compete with the colours in your countertop

I think either can work depending on what look you’re going for and what you’re trying to highlight. My only thought would be a lighter grout behind a stove will get constantly dirty and will be harder to keep looking fresh than a darker grout which may be a little more forgiving.

If you use an epoxy grout the light color will stay light.

I vote for all of your lights to be the same color/temperature.  The range is warm and the undercabinet lights are too cool. Pick the same for everything!   
As for grout, something close to the tile color would be best.  
Good luck!

Noted, the lights are strip LED, I had unplugged them a bunch during the tile install so I didnt get a chance to reopen the app, link them and reset the temps. Thanks tho, this does bother me in the pictures too.

Grout with a creamy colour close to the tile colour otherwise your backsplash will look too busy. You can’t look only at the backsplash, you must look at how everything looks together. Also go with a warmer light 2700 because you have warm tones in your kitchen. The other light you have makes everything look pink - totally wrong.