That unit has reached the end of its life, I'm afraid, and should be replaced. Better to do it now, while it's still "exposed", and before you go to the expense and trouble of adding a chimney cap and facing material, than prolong the inevitable. As for determining the manufacturer, there should be some sort of metal identification placard on it, perhaps behind the hood. If there is it's likely covered in soot, which is making it hard to find. But even if you are able to determine make and model, I highly doubt you'd be able to buy replacement parts and/or accessories, OEM or aftermarket; it's just too old.

I do think those two trims have something in common. Simplicity. Neither one has mitered corners, which would be a tad more time consuming than simply plopping in a rosette or abutting the top piece against the two uprights. It would not bother me to see both in a house that has stood the years. They are probably both sturdy and have lasted the years because they are well done. That is more satisfying than woodwork matching throughout, I think.

When was the house itself built? The interior design aspects are on point with the mid 1980s but the architecture is earlier mid century. I can't really picture someone building a house in that style and decorating aspects of it in late Modernism/post Modernism at the same time. I can picture someone updating a 1960-odd house this way.

Hi! Cute place. Wonder where the toilet and mechanicals are :-) Hard time reading dimensions for me. so. First I would remove the doors to the kitchen and to the living room that will help. When I think of sight lines and such opening that space from the living to kitchen - you would be looking mostly at the side of a cabinet. I would not value that. Widen the doorway from living to dining and then dining to kitchen. Your suggestion of the fridge.. it would be a great use of the depth and bonus!! right near the dining room which is ideal. In a perfect world and this was my space I would be inclined to remove the stuff from the middle of the plan - that hall (flip next to the bedroom closet) and dining closet and widen the kitchen to dining walkway.