My advice for the paint & corners:
Go off white, pick a light color from the natural stone wall or a light sandy off white and create an upbeat, more visually open space. Whites give you a lot of options for accent colors throughout the rooms that can be easily interchanged if you're tired of them without having to do something big like repaint again. Start with the big things, like the wall paint, then figure out the details, like the furniture. I also recommend big, oversized plants like a palm or fig tree because you have a very laid back, natural vibe to the space with the wall already.

https://www.houzz.com/photos/contemporary-tall-indoor-palm-phbr0lbl-bl~l_51677

If you're considering art I have a few fun and bright options!

www.etsy.com/shop/laurenelysefineart

Good luck! :) & remember, start with the big, ease into the small.

1. Dining room furniture - should it be dark like our island or light like our cupboards? Should we just purchase one large dining room set or a smaller one and a sidebar?

A:Looking at your white set, I think darker is definitely better! The space is very small so if you decide to buy a new set, I would get a round table with a leaf or two, that way it will become oval when needed, but nice and compact when you aren't using it. No larger than 56" diameter. A small sideboard that is low would look nice under the window, with nice silvery grey thai silk curtains from Pottery Barn to complement the space. The grey will tie in nicely with the brick.

2. What colour should rugs be in both rooms and should they be the same design? The back wall is brick and is throwing me off.
A: I wouldn't bother with a rug in the dining room, it will make the space feel small. If you wish for a rug in the other room, try and tie in the colours with any existing colours. If you wish for simplicity, Sizal rugs are a safe bet.

3. What colour should drapes behind the couch be?
A: Silvery grey, thai silk is beautiful. elegant and classic. If that's out of your budget, then look for knock offs. Do not get patterns, it will 'fight' with the brick.

4. What do we hang on the walls to tie both rooms together? A large piece of artwork above couch? A row of smaller pieces from dining room to living room? Mirror? \
A: Two large pieces of artwork that tie into one another would be great. Something on the traditional or contemporary side will work, a splash of colour would be great too. Add colour to the room by buying new pillows that will either tie in with curtains or whatever rug you decide to buy.

5. Our t.v. is on a t.v. stand right now. Should we have a custom built one and if so should it take up the whole wall.
A: Do you have a fireplace? If not, I would buy a gorgeous electric fireplace and hang the TV on the wall above. Keep in mind, you do not want a high fireplace or you neck will hurt when watching TV. They also have TV stands with build in electric fireplaces that add warmth to the room.

6. I always have problems filling up the corners. Any suggestions? Lamps, plants? Shelving.
A: Your room looks very cramped for space, I wouldn't add much more, but a small accent table or lamp from TJMaxx or Homegoods is something I use for my small spaces. The plant beside the TV stand currently looks out of place, if you had a plant stand for it, then in you could place it in a corner if there's space. Your dining room is small, I wouldn't put another else in it. You can 'stage' your sideboard with trinkets once you've purchased one. Please measure the sideboard that it's not too wide otherwise the curtains cannot go to the ground. Good luck shopping.

Thank you so much for all of your suggestions. I especially like the idea of a gorgeous fireplace with the TV above it.

My pleasure. Please post pictures when you've gone shopping.

perhaps your 'cozy' would be assisted by adjusting your furniture placement -- lined against the wall always reminds me of a wake or a bridal shower. you didn't mention much about your taste.... the brick wall and dark cabinets would look amazing with a contemporary 'loft vibe' , flanking the lovely bay window with facing sofas would lend a more formal feel, a dark [storage?] banquette bench on the brick wall could offer a more casual feel and also offer extra seating and an option for dining table placement. [which should not be white:( sorry. dark like your cabs. is good. the dark bench would also continue the lines of your kitchen cabinets.