Paint the entire post the same color as you paint the walls. I think a softer warm light neutral would look better than the colorful scheme you have now and you would not have to paint the brick, which is attractive.

I don't understand your comment about matte. Matte is a flat finish that is scrub-able. It should be perfect for kids. Satin and other sheen's will show imperfections in the walls. If you really don't want to do matte, do eggshell.

Paint the beams the wall color including the crown that is now white. Leave the baseboard white to match rest of room. Leave brick as is. Paint FP screen black. Match the hall paint for the LR & DR

I wouldn’t hesitate to paint everything white. Make sure you use the proper prep and product on brick.
Get a much larger rug. Matte finish on walls and satin finish on baseboard and other trims all same color. It will be amazing. Many off white choices available. Just get 3-5 color samples from samplize.com to help choose best tone for the natural light in your room. Here are a couple inspirations for you.

Then put large art over your fireplace. The mirror doesn’t reflect anything special so art would be great!

To maintain a 70's air, please don't paint the brick. I'd paint the dividing wall, beams, and posts the same as the ceiling.