I love your trim and I hope you'll keep it stained. Your beautiful home has a lot of character. These are a few suggestions. Take them or leave them since ultimately it's your home! Replace the traditional / fussy drapes with something simple. Replace the dining room chairs with something with clean, simple lines. Is there any chance your TV could fit inside of one of the built-ins? If you removed the bottom shelf, the TV would be much less obtrusive tucked in there. Then you could put a nice piece of framed art above the fireplace. Remove all of the small decorative items from the surfaces: buffet, fireplace mantle, shelves. Replace with fewer more impactful items. Remove one of the cabinets from the dining room. Right now you have three and they seem to be differing styles. I'd be inclined to remove the corner one since it seems to coordinate less well with the other items. Replace the slipcovered sofa with something new and more tailored.

There are some truly amazing options out there for hidden TVs. If you were open to any of them, that might make your arrangement easier. being able to place things without having an obvious TV in the room when it isn't in use. We have a TV mounted above a fireplace and it ends up straining your neck to look up that high from a couch. Keep the TV low wherever you end up with it.

If you don't want hardwood in the kitchen there is no reason that you need to do hardwood in the kitchen. The current trend is to have one flooring throughout your home. Prior to the last 10 years it was much more common to have different flooring in different rooms. Usually a tile or laminate in kitchen and baths, hardwood or carpet in other living spaces. Trends come and go. Pick the flooring that will best fit your lifestyle and make your home work for you. There are many ways to transition from one flooring to another. The transition is easy when there is an architectural break between two room (doorways are most common). Sometimes we can use flooring to define an area and create two spaces where there is no architectural break. My last home didn't have an architectural break between the living room and the entry, but the transition from carpet to tile defined the areas. Here are a few other transitions that are obviously a design choice where two floorings were intentionally selected.

No I don't think you need to get a shorter table unless the one you have is wider than usual (most tables are 3-3.5' wide). I don't think the length is the issue in that room, it's the width. You can just get a bench instead of chairs on that side. Your table is probably 6' long or so, right? Then it will still fit in the dining room. You may want to play around with just putting chairs on 2 sides instead of 3 and pushing it more towards the corner to avoid encroaching on the living room space. 6'+3' = 9' so you'd have 1' to spare. If you want to share some space with the living room you could put seating on both sides.