Could you build some closets between it and the entry and still leave the remainder set up as a living area, or would that make it too small?

It's hard to picture, but I think not. There is only a small open door way, and then most of the front wall is a window.

I will look though - maybe the side wall could somehow be enclosed. Will look into that.

Maybe you could do one whole wall in bookshelves and storage cabinets?

It could look like a library/living room. If you put in some full length cabinets you could use as closets, you would have a place to put coats, shoes, and things you want to keep away from your dog. Take a look at Ikea cabinets-- you could configure a whole wall any way you wanted to.

This is a pin of a mudroom storage created from items from Ikea -- might work along one wall .....

..... Then add a simple table and a couple of armchairs -- for reading the paper in the sunshine ...

Here is a link that might be useful: Pinterest -- mudroom storage DIY Ikea

Here's another idea to add bookcases (you could add spaces for stashing stuff too) .... a table and a couple of comfy chairs .....

Here is a link that might be useful: Pinterest -- another possible idea

Would a wardrobe armoire work to store coats, bags and shoes. I use one as an extra closet in my master bedroom and one as a game closet in my formal living room. They can house alot of coats and usually come with strong hanging bars.

Is it possible to add storage under the stairs (built in cupboards and/or drawers)?

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I'm liking the armoire idea. I think I could find one to complement the dining room furniture. A sturdy one - or 2 would cover a lot of the mess!
The under stair storage won't work there because that wall is where I have a large dresser type storage.
Thanks for the ideas so far.