DAP makes a good patching compound. Stainable, sandable, paintable, and hard as rock. I did the same thing on my garage door, and some of my trim around my storm doors where the old holes were still visible from an older storm door, and painted over it. Good as new. But, I agree, you'd have to paint the door if you want something pleasing to the eye.

I'ld replace that entire board ... looks to be a 1"x6" with a rounded over edge. Get it in a pre-treated pine variety of the appropriate length and paint it white to match, and fasten it securely to the wall. That looks like moisture damage, most likely from either warm air coming in through the cracks and leaving condensed water on your wood, or cold air coming in through the cracks and condensing the moisture in the house where it ends up on the wood. Either way it's bad for your wood. After you have replaced the board properly, you will definitely want to look in to replacing the weatherstripping and seals around that area to solve the actual cause of this problem, or you will end up replacing the board again in the future. Pre-treated lumber will stand up to that kind of abuse for longer than untreated wood, but in the end, the water will win.

I planned on SW Collonade Grey for the walls/ceiling, a “custom” blend walnut stain on my red oak floors and the same stain on the Alder beams above. Mantel will be the same walnut stain too. Sounds lime a lot of stain, but the kitchen is tile floor with white oak cabinets. What stage is your build - can finish selections be changed at this point? If so, I would recommend a consulation with Maria Killam (https://www.mariakillam.com) to ensure your finishes will be cohesive and provide the proper background for the overall look and feel you want to acheive in your new home. Your fireplace is fine, but will be a bossy element to work with (start reading Maria's blog posts and you'll see what I mean).

Remove that thin, tan strip and put in a wider piece that covers everything. If this is outside your skill set, any competent handyman should be able to do it.