I think whomever did this, did not have much budget to work with. A hood is there. Use it or consider to get that 99 dollar drop down installed at the correct height, and remove some upper storage as similar to the idea regarding cabinets posted above here. That said, if you have to hire it installed, cabinets removed, dry wall repaired(or just cover the holes up with a flat piece of stainless or even pressed tin ceiling tiles all the way to the top of the existing cabinets(they are not expensive)used as a backsplash behind your 99 drop down hood.....<https://www.americantinceilings.com/f1/>,  the stainless version. If you are on an inside wall, and have to get code inspected etcetc in the city...probably still going to cost you 1500 I would expect in my area.... IF it is vented thru the roof, and you will want it to be, truly! Maybe you will get by with less, just know that making it vent thru your roof, if the existing one does not, will cost more than that possibly... and than there is the roofing /new water leak potential. No code inspector would allow that installation you have, imho, so maybe you are in a rural area. If you hire someone to help accomplish this, make sure you hire someone who can address the drywall, electrical, venting, roofing and tin ceiling tile as a backsplash.(An updated home I know of used that as a backsplash and it looks very nice with the gray tones of the counter and the black kitchen appliances. There are other stamped patterns in this price range, so you can find something you like better perhaps?) Otherwise you are looking at a bevy of home improvement people, but I think it will look much more appropriate and be an asset to your kitchen when done. Just really make sure you get someone who has a brick and mortar store or a local excellent reputation, not a 'chuck with a truck' from "won't be in business next month", and don't pay them in advance for their work either. I hope you can find a way to make it look nice and work better also.

When moisture and grease particulate laden air flows through a duct, there is turbulence and these two components will be brought in contact with the duct wall. Slow moving air will allow condensation; fast moving air will allow impingement collection. In general, both likely happen at different ratios depending on temperature and fan setting. I wouldn't change a duct diameter just for this reason. What you have for a hood is unconventional. Commercial hoods and "pro" residential hoods use baffles that cause two 180-degree turns in the air flow. The larger grease particulates are deliberately impinged on the baffle surfaces, and given some baffle angle will drain toward a collection point. Even without angle, the grease is trapped where it can't easily drip down onto the cooktop. I don't know the details of this fan scheme, so I won't comment on what it might be doing with grease. When you initially wrote that you were having the hood installed, I didn't bother with commenting on its design -- sorry. Whether grease dripping from the fan grid was intended or not I cannot say, but it likely is consistent with the physics of the hood design. Again let me pontificate for this forum that the farther a residential hood deviates from long-time optimized commercial hood design, some portion of performance and efficiency will be lost. Running the fans for a minute before generating any greasy effluent might help with condensation, but I think the fan blades here are causing grease collection around the grill as well as wherever it is supposed to collect.

It is actually desirable that the hood capture area be extend some distance from the wall, preferably more than 20 inches, such that it overlaps plumes generated from front burners as well as from back burners. Front-to-back limits due to sight lines to back burner pots need to be considered. An example is the Wolf Pro Low-profile hood that is 22-inches front-to-back, probably 20 inches of effective capture area. It is 10 inches high, so one might want to jack the cabinet over it a bit higher to get 30 to 36 inches of clearance. Spaced out 3 inches for a smooth transition to the cabinet doors one would have very good overlap in the forward direction.

I have purchased an under the cabinet model that vents out from behind the unit through an exterior wall. i assume it attaches to the wall at the back and or on top to the ceiling. so I assume to be safely attached to the ceiling the prrcut holes in the unit ehen in position need to align with ceiling joists. or could a thick piece of wood the shape of the top of the unit be secured in place to the ceiling to simulate the bottom of a cabinet allowing a secure fit?