Yes, those swinging doors make it a challenge to get to those shoes, don't they? If there wasn't room in the closet for the cabinet, I would consider getting one of those over the door apparatuses that would hold the shoes. I have similar awkward spaces in my home; they should make the designers have to live with a home for six months to understand how awkward things can be!

It's such a tiny space there, I would not put anything there.

If you really need to have storage for the shoes, I would consider having a custom corner unit built so that it is out of the way when the doors are open. This way you still have open shelving to quickly place the shoes in.

To this day I don't understand the shoes by the door thing unless it's a back door. Why not in a closet? i vote for leaving the space blank.

No measurements for your closet. See if this 3 in 1 can fit inside it.

A small corner table would work but harder to find the right size. If you could find a small round drink table, those would work just a well and just have it a nice small entry piece. Good luck.

I would store the shoe cabinet in the closet and not put anything there; if you have to have something, maybe a slender hall tree but they can look cluttered to so I'd go for nothing if possible