I think Anew Gray it is good combination with your floor.

If you decide not to paint the brick, can you paint the back of the bookcase a color similar to that of the darkest brick, and the rest of the bookcase to match the wall or a shade darker?

I don't think grey would go well with your floors. I wouldn't paint the brick but I would paint the built-ins and wall surrounding them a light color in the brick or flooring.

Here is painting the bookcase Anew Gray and fireplace Tricorn Black

White over the fireplace. I think I like it better natural and just the bookcase painted

Are you keeping your stained wood trim around the baseboards and windows? If you're painting it white, then paint the shelving the same white color and keep the brick. If you're keeping the wood trim, you need a warm or neutral white for the walls, and can then use deeper shade, more of a taupe or beige, on the shelving. Anew Gray doesn't work with your floors or brick.

Thank you for everyone's comments. We pushed ahead with anew gray, since we were halfway through painting anyway. I am curious as to what program or app you use (specifically @Melissa R) to do a color mock up of the photos...

I sort of like the color now that it's on the walls...or at least okay with it! We also did a hallway in that color and with the same flooring. I'll post a few "after" photos when the furniture and whatnot are all back in place.

@User Sherwin Williams site has the program!