What should I do with these bricks?

Paint a light tan or vibrant red to update your yard and home . Black is also common but I prefer reds , tans , to keep it modern

prime with masonry paint an then try a small area with neutral gray or mocha (very light beige) color pallet to be matched to wall color.. Once painted (a bit of a tedious task) the brick can be dusted or wiped with damp cloth to keep clean.

I found a Better Homes and Gardens article about "Exterior Color Combinations Done Right"
https://www.bhg.com/home-improvement/exteriors/curb-appeal/exterior-color-combinations/

How about a stressed white?

If you want your fireplace to stand out as the focal point of a room, choose a paint color that contrasts with your walls to help highlight it. Light and dark colors create an easy contrast, so if the walls are pale yellow, a rich terracotta paint can work well for the fireplace.

It depends on the rest of your decoration of course and if you want to blend in the whole mantel or just a part of it. You are not obliged to take the same colour for the whole thing. You could paint everthing, except behind the wood stove until the shelf and the floor where the stove stands on, in the same colour as your walls. and choose something else for the rest.

Can these bricks be cleaned? Are the dark marks the result of soot?

Definitely scrub the brick and mortar prior to painting, to get better adhesion.

I personally really like the look of brick. I have a fireplace made of brick. After many years, they had became darkened from age and from burning the fireplace. I went online and found several products especially made for cleaning bricks. I bought it, used it and the fireplace still has a used look but it is really beautiful. I'm very happy that I decided to clean the bricks instead of painting them. You will have to make your own decision. Good Luck!