What is happening in this part of the house? Is it usable space for the kitchen? It looks like it may be a closet behind there

Not sure if that space is usable for the kitchen—I think the utility closet (furnace, breaker box etc.) extends from that closet door all the way through that space.

Is it possible for you to change the refrigerator door so it matches your cabinetry?Then the refrig will look more like a closet or cabinet.

I don't think it's so bad or so awkward. I'd live in it and make a plan as suggested above.

The worst thing to me is the lack of a full sized sink (it looks like the one in the peninsula is on the small size almost more like a prep sink), why it's not under the window instead is a mystery! Plus the refrigerator is a little far, it would make more sense to not have the sink on the island so you have a landing spot for the fridge. However I've seen much worse.

Does the peninsula countertop impede that louvered door?

I love the long counter top run you have under the window. We arranged our kitchen to have that and it’s great for cooking that has a lot of things going on, making pasta, or making cookies. If you don’t need it and end up rearranging, you could consider a large sink under the window, the fridge where the stove is and putting the stove where the sink is located. Moving all that stuff might be pricey with plumbing, electrical, etc, but live with it for a year and then decide.

It looks like the kitchen is probably not original to the house. If that's the case, it's possible a previous owner tried to improve it (most likely to make it bigger, and open it up to the living space) resulting in the layout you see today. I agree that living with it for at least a year will help you to understand the merits and flaws in the current setup so you can make good decisions for the next iteration.

That’s tough! I found that having a fridge in the walkway was very difficult when more than just our family was in the house. When more than a few people came over it became a bottleneck.
Unless there’s something else that you love about the house that’s not possible to get elsewhere or just as easily: neighborhood/schools/commute/view...I’d pass on this. It’ll cost a lot more than you think to do a remodel. If you absolutely love the house and don’t mind putting in money/time to make the kitchen exactly what you want, the go for it! Maybe someone will have some ideas on how to change the layout.

A wall placed opposite range would give you a U shape kitchen and option of cab depth fridge placed against that new wall. But dimensions would be critical . Sink to window position allows the chunky sink peninsula to be cut back and just extend counter to right of range around the jog up to louvered door. such major work however w an old building : I’d think if maybe these things won’t be done??? for reasons valid, you may yet discover .....can I live w existing situation??? Is foot traffic in the entry really going to interfere w fridge use? Seems your major complaint. For row house I think it’s oversized kitchen and would want it more compact and more living space instead. Fridge in that spot reflects the bigger open kitchen they wanted w light flowing . It has some positives And you would not start doing things for a yr. and before that, think if the expense is worth it. Has the price come down?

Since you mention you are redoing kitchen, have you considered moving it from there? Can you flip the spaces?

The only thing I don't like about it is you can see it from the front door !