It won't be a full sized closet, but yes....take advantage of that space! I will use my closet under the stairs to store Christmas decorations that I don't want exposed to the heat or cold of the attic. You'll probably need to get a custom closet door made for it since a full height/width door won't fit.

The reason we did the coffered ceilings was because we have 12 foot ceilings, and cabinets all the way to the ceiling would have been cost-prohibitive because it would have needed to be custom instead of semi-custom. I think you could definitely do it with 9 foot ceilings. The beams would just have to be shallower. In fact, our inspiration kitchen, in a kitchen showroom, had much shallower beams than we have in our kitchen. I will put a link below. Scroll down to the last picture in media, and click to enlarge. That is the showroom kitchen for this KD, who gave us the idea to bring out ceiling down by using the boxed beams. I would think that while it would be easier to install the beams before the cabinets were in, it could be done after. Not sure, though. Here is a link that might be useful: Kitchen photos with coffered ceiling

I just went and measured my (unremodeled) kitchen, it has 17" clearance and I never thought of it as exceptionally low, it looks 'normal' to me, but maybe just because it's what I'm used to. It's certainly never impeded my counter use. Are you going to take down the drywall on the ceiling to put up the coffers and planks? That'd buy a smidge more height?

I am not real excited about the random sized coffers in the row of the table, or the wide leftover piece by the bathroom I think I would try something like below, at least correcting the area around the bathroom or in the other drawing having the coffered ceiling extend over built ins that stop short of the ceiling and see if you can get them all the same size in at least one dimension. The jog next to the table is very awkward and has the built ins "under" a beam at the front and to the edge of a beam on the side.