Yes, you can totally do this--make sure you use a starlight glass tile since the regular tile will look slightly green. Did you see the white glass tile that Kaidan is looking at?

Without seeing the slab it's hard to know for sure but here's one from Jeffrey Court's Antico Portugese line color Country Gray: These are kinda cool. I know nothing about the pricing. These are cement mosaic tiles by Villa Lagoon Tile. The colors are complements to Sherman Williams colors. I like the Pearl Gray below. It's a match to SW Mindful Gray. Last one for now, a more traditional 6x6 Italian Tile: This was from http://www.ceramic-tile4u.com/ . Budget tile too at $4.04 sq. ft. It was their Arpa Star tile in gray. That site has HUNDREDS of tiles to look at. May be WAY time consuming but worth a look. Good luck!

"is there an option to remove the 4" countertop backsplash?" It looks like granite from the photo, so there should be a seam at the angle where the backsplash meets the top of the counter. In which case it can be removed without taking off the whole thing. Using a bold color would help liven up the space. Something that accents the colors in the countertop. Then find a backsplash that works with them both. Or maybe something neutral in a texture tile to give it interest without add too many colors. Change out the hardware to match the color of new appliances, and consider changing the pendant lights to a modern fixture that matches as well. Best of luck!

I think doing a mix of different white tiles, or a mix of gloss and matte tiles might be easier to harmonize with the other two whites than having an expanse of a single white in the backsplash. These ideas are from Heath Ceramics. A mix of Heath's canvas white and vanilla bean. I don't know if I like this pattern for you, but I like the color mix. This is a dual-glazed tile; part of the tile is glossy and part is matte (I don't know if anyone but Heath does this, but you could mix gloss and matte tiles from someone else for a similar effect)