Effectiveness is influenced by capacity of unit. If you are trying to mount a 270 cfm fan more than 30" above cooktop, then it may make noise but will do little to clear the air. If it is 1000 cfm, then it can be higher than 30". So the answer is "it depends". Most people don't mount them more than 36" above the cooktop unless it is a really high powered unit.

So yes eliminate that section between living room and kitchen. Move your living room furniture forward closer to tv. Now you can have a separate dining space where living room furniture used to be. You may have room for a kitchen island with cooktop. Then sink where stove was and refrigerator where sink is now. Double wall oven where refrigerator used to be.

I would consider putting cabinets on the short wall where the utility wall is. Have your stove there centered on the area. Move the sink so it is centered on the long wall where the cooktop is now. Put the fridge on the end run of cabinets closest to the dining table. That will give you an L shape kitchen with a nice work triangle. If you have your heart set on an island remove all the wall cabinets and install a large island. This is not the style of your kitchen but it does show the layout I described above. L shape cabinetry with stove on short end, sink in the middle with DW to right of sink with cabinet above it for dish storage and the refridgerator on the end run so it is out of work triangle and conveniet to the dining area for serving or work triangle for prep and cooking. Island can be any style you desire. L shape with island and open across from island; you might consider this layout if you desire an island. Something like this?

Totally doable as long as the island is not too bulky. And of course if the circulation space around the island is at least 3'-0" wide. You don't need too much space to move around in a kitchen because you should be able to pick up a hot dish and put it down on another surface without having to move too much. 12'x14' is a large size and a work island would possibly make the space more practical to use. In the pictures attached, the work island is only 2' wide but 4' feet long