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Start detective work. Begin with the fan motor itself. Is it balanced, or does it wobble? then look at the housing. Caulk and insulate anything that moves when it's not supposed to. Then the duct cover....and keep going. You may not eliminate noise, but if you eliminate every little bit of noise from every source, you end up making a big difference overall. One of the bigger possible culprits besides the motor itself is the duct run itself. It should be secured to studs and have rubber isolation strips between it and the studs. It should be joined together fully with mastic at every joint. It should have an insulation blanket around it, and it should be encased in a drywall chase which has either foam or blown in cellulose.

In looking at your hand drawn diagram with your revised plan, I think kgwlisa's suggestion is the one to use. If you move the FR door that is in the kitchen, you'll have a very functional layout. That would mean to shove the cabinet section that contains your cooktop/oven down to join up with the other wall of cabinets. Then use the space you have beside the relocated pantry to put the doorway into the family room there. I don't think you'd need a door on it, just leave it as a doorway into the FR. If the other FR doorway is too close, you can consider it a column and add moldings if desired to enhance that look. You might feel like why have two doorways so close together when both lead into the FR? It's because one goes to the walkway you'll create along the kitchen and one goes to the LR. I take it you'll be using the upper end of the kitchen room where the window is in your first bird's eye view as an eating nook, since the doors to the basement and mudroom would work better there. Perhaps you can even enlarge the window there to provide more light into the kitchen? JMHO of course without knowing your budget, etc. I think you'll have a kitchen you love!

Could you please post a larger version of the Kitchen and identify what each item is & the size? Is your sink really 66" wide and/or 2 sinks (it looks like two in the "picture" of the Kitchen)? If so, could you please explain why? Do you really need two DWs? Is this supposed to be a Kosher kitchen? Is that a table where it says "Family Room"? If so, do you really need two forms of seating so close to each other? It looks like you have a Dining Room as well, is that correct? Unless you have very long arms, your island should not be any deeper than 60". Most people can easily reach no more than 30" without straining, so anything deeper will be difficult to clean (reaching the middle of the island). Keep in mind the following for seating at an island (or peninsula): Each seat needs a minimum of 24" of linear space. So, to seat four people, your island will need to be at least 8' wide (96"). The minimum recommended seating overhang for short to average height people is 15" -- after accounting for any doors or decorative end panels on the cabinets under the overhang. (If you are a tall family, an 18" overhang would be better.) For cabinets under a seating overhang, I would not go any deeper than 18", and 12" to 15" would be better. Why? Because you have to squat/lean over & under to reach things inside a cabinet under an overhang and you will have trouble reaching into the depths of a cabinet much more than 15" or so deep. No drawer in the top half of those cabinets as well as the overhang will block access to the drawers from the top. Your island would be better as: 1.5" overhang in the front + 24"D cabinets + 15"D cabinets facing under the overhang + 1" for the cabinet doors (doors & drawer fronts are not included in cabinet measurements & they stick out another 1" or so from the front of the cabinets) + 15" overhang ======== 56.5" When considering the width of your island, be careful not to make the aisles around the island too shallow. Plan for 48-inch aisles around the three non-seating sides and 60" b/w the seating side and the table in the "Family Room". (Aisles are measured to/from the items that stick out into the aisle the farthest -- counter overhangs, appliance handles, etc. They are not measured to/from cabinets because, as I mentioned above, cabinet doors & drawer fronts are not included in the depths of cabinets. 24" is the depth of the cabinet box only, not the doors/drawer fronts. Counter overhangs aren't included in cabinet measurements either.)