Personally, I would never remove wainscoting. It gives a room character. I don't think I've ever seen a ceiling fan in a DR. I would replace it with a chandelier and center the table under it. If space is a real issue, you could consider a drop leaf table.

You could go with a deeper color on the walls and keep the wainscotting white. If you did that, you might choose some full curtain panels in a color lighter than the walls. I would definitely add a chandy instead of fan. A nice rug would help ground the space.

Are you looking to buy a new table? One that is more narrow?

CV, my husband's office is painted SW Pewter Green and it is gorgeous!

I do not have a pic but here is a good one. It looks more green in real life.

Measure it out. You might be able to either.

Here is a round table.

Also, check out your local Craigslist. I have purchased two round tables with chairs for a steal and loved them! One set I painted and the other was wood which was perfect.

I think an egg shell color would be nice for your walls, not too white but it will lighten the room. Personally I would get rid of the ceiling fan too and have air conditioning put in.

I like your house. You are so lucky to have wainscoting!

What about some sort of hidden ceiling fan? Not necessarily this style:

We had the same sun/heat problem with sliding doors in our kitchen. We had a 3M sun control window film installed and it helped a good bit.

I get that sometimes you have to choose comfort over design. Even with air conditioning, there always seems to be one hot room in a house.