To me, shutters look dumb when they can't possibly cover the windows that they flank. Painting brick will be turning a low- maintenance exterior surface into a relatively high high-maintenance surface. That is not what I would be wishing for on my house. Apart than the shutters, it is nice-looking house from the street. I would not mess with it for a while. The interior shots make it look like some nice spaces too. If I owned it, I would be jumping around like your girls are.

The word you may be looking for is efflorescence, a white powdery scum that appears on masonry. But the placement of the spots makes me think it's not efflorescence. I would leave the shutters on. Removing them is going to expose holes where they were attached and shadows where they once were. There are lots of ways you could go with this. I would paint the vinyl (hardiplank?) gray to match the mortar between the bricks, paint the shutters and arches above the upper windows black, and for the door, either leave it as is, or a teal color, or an autumnal orange that goes well with the bricks (assuming I'm seeing the color of the bricks accurately). Since you just bought the house, live with it as is for a while and look around the neighborhood at what other people have done.

I like a brick color with a grey stone. Around here that typically means granite for the stone since it is supplied locally, but a bluestone or a gray cement would work well. Just make sure that it works well color-wise with your red and gray pavers as well as with the house color. I would only use two masonry materials in such a small space, though if you want your fire pit to be a freestanding metal piece, I don’t think that will be a conflict.

I would do the same as the house or at least just the frontage .