I love spending money (ha) but I think spending $20 for those RH knobs might be a bit of a waste - they aren't distinctive looking, and I think you can get similar ones elsewhere for less (Top Knobs is a great source - there are others).
I normally hate painting wood trim, but I think you'd be ok to paint it if it bothers you. Right now, it coordinates pretty nicely with the counters and backsplash, but it does feel out of place style-wise next to the beefed up molding on your cabinets. That said, it isn't especially distracting either - you could leave it as-is, especially if you add additional window coverings/treatments.
If it were me, I'd probably have gone with a woven natural shade here, but what you've done is neutral and works fine. I don't know that I'd go too fru-fru with the valance. Maybe consider a simple Roman shade in a flax/neutral linen? I'm torn because a) I know how much window treatments cost and b) roller shades are really practical but the style of your kitchen almost needs a softer/more organic window treatment to break up all of the hard lines in your space.
I agree with the recommendation to splurge for new barstools. No painting - put those on FB Marketplace and be done with it!
As for the ceiling fan - do you use it to cool down the room? If so, keep it. If not (we're in Southern California and would never really use one), replace it with a nice island fixture (one that provides ample light for working in a kitchen but that also works with/not against the formal feel of your space). I could see a single bar-type island fixture working quite well (single pendant with multiple lights on it in a row) - maybe something in an unlaquered brass or even copper to break up all of the dark colors?
I would NOT change out the range hood for stainless, but I would consider painting it in a tone that matches the cabinets (or if you replace the backsplash, in a tone that coordinates/pulls from that selection) so that it looks more cohesive/intentional.
If you are considering changing the backsplash, I could see something like this - although ,you should make a selection in terms of color and style that works in the actual space (just giving you an idea) - I think a tumbled/rough limestone type tile in a creamy tone could work well with the countertops and tile.
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