I want to echo Weed's caution about being sure on code compliance when having your vent going out near that window. Last year, when I was doing a re-routed of the vent on my range hood, the local building inspector happened by. (We are on pretty good terms so there was no compliance issue.) I asked him about the rules and got a long dissertation as he handed vent tubing up to me. The gist of it was that the rules for range hood vents are different than for those that directly vent gas appliances (say a water heater power vent). There is some discretion when it comes to venting from a range hood over a gas stove. How far the vent has to be from an opening window also varies with whether the vent is above the opening window, at window level or below window level. I was told the vent that is above an opening window can be closer than one at or below window level. BUT, you want to be sure that, if you open the window while the hood is running, it won't be sucking in the vaporous stuff that was just blown out the vent.

Thanks, all, this is useful: cawaps: full-width duct cover, that'd be perfect! I imagined my contractor rigging something, but of course it makes sense they make hoods that come with such a cover built in. huango and palimpest: the straight regular chimney does look nice, and if space/construction allows that'd be great. In our case it's not a full reno, though just the hood, so if breaking into the ceiling adds much $$$ that might not be worth it. But if it isn't a big deal for the contractor, we'll do it bellsmom: I get what you're saying, and yes, that would work too if for whatever reason a hood with full-width cover isn't a good option. And it would give us a bit of storage, too. Problem #2 that just ocurred to me is the side wall exhaust would come out immediately under our retractable deck awning. Hot exhaust under vinyl (or whatever it is) might not be a good idea, or at the least it could trap the smoke right there by the patio door. Hmm. If our fridge and stove were swapped this could work. In the fridge location, the stove would back to an exterior wall with no utilities or external obstructions, and the little over-fridge cabinet would be the right height for an u/c hood. Would have to remove the over-stove cabinet to make room for fridge, but no big loss. But, this would mean running a new gas line and a new water line (ice maker), and leave the stove closer to the sink than we'd like. Oh, how quickly it all snowballs!

The assessment of trades above by @User is correct. You’ll need a HVAC person, an electrician, and someone to do drywall. You’ll also need a new dedicated electrical outlet for where you moved your MW. I’m not sure why there isn’t an outlet above the range, where you moved it from, but it wouldn’t be in the right place for a chimney hood if it were. That’s why you have to decide on the hood, and design everything around the exact hood’s specs. The location and shape of the duct and the electrical location is NOT universal. It’s normal to have to move or adjust everything, or do new. You need the specs and the hood on hand when talking to contractors. Don’t forget an exterior damper to keep cold air from infiltrating your home! You will get a better job, done faster, if you hire a General Contractor, rather than trying to hire the trades individually. They have their subs that they give lots of work to, and who will return phone calls and show up. And pull the permits that this will require. A homeowner doesn’t get the same price or response from the trades, and it takes weeks longer. I’d suggest posting on your local NextDoor or Facebook neighborhood or city group to look for a licensed and insured (check this!) contractor who can do a small job. But, it isn’t as small as you think!

To be even slightly informed about this subject, you would have to read quite a few hood and make-up air (MUA) related threads on this forum. We could also point you to commercial literature that one might extrapolate from. I can provide a somewhat standard spiel touching on the highlights. I cannot provide reviews of hoods I haven't owned. There are two maxims that one might keep in mind. (a) Performance, aesthetics, affordable cost -- pick two. (b) The most efficient hoods (cooking plume capture and containment performance per watt of electricity usage) are commercial hoods. For an idea of commercial hood design and function, read the first dozen pages of the Greenheck Guide, available from: https://www.tagengineering.ca/wp-content/uploads/2015/02/KVSApplDesign_catalog.pdf. What a hood system is expected to accomplish is capture and containment. Capture is effected by having a hood canopy that overlaps the rising and expanding cooking effluent plumes so that these plumes pass into the hood volume, preferably under their own power. Containment is effected by pulling the plumes through the hood filter system -- preferably baffles for most serious hoods -- and thence via the ducting to the outside. Containment requires adequate volumetric flow rate (CFM in the USA). Flow rate achieved depends on the blower selected and the pressure losses moving air through the baffles, through the hood, through the ducting, and through the roof or wall cap to outside, and then, most importantly, the pressure losses in getting fresh outside air back into the kitchen (MUA). Flow rate required depends on the velocity of the cooking plumes and the area of the hood entry aperture. Obviously, one wants the achieved flow rate to equal the required flow rate. But wait, there's more: The specified flow rate for blowers is that achieved at zero pressure loss. The flow rate at some pressure loss is less, just like strangling a water hose decreases water flow. One has to specify a blower strong enough that for the conditions present the desired flow rate can be achieved. (Not otherwise addressed here are the desirable hood features of lighting and blower control.) Fortunately, in many cases we can apply rules of thumb to the selection criteria. Some arithmetic is required. Overlap the hood by at least 3 inches relative to the cooktop or range top on all sides not including a wall. For a 24 inch [front-to-back] range, achieving a 27 inch wall hood may be difficult or interfere with sight lines or head position; in such a case a 21 or 24 inch hood front-to-back depth may do with some plume escape possible. Calculate the area of the hood aperture in square feet and multiply by 90 ft/min to find desired CFM. Multiply that result by 1.5 to roughly find a specified CFM that should work. Choose a duct diameter per the hood installation directions, but the goal is an air velocity in the duct between 1000 and 2000 ft/min. This equals the desired CFM divided by the duct sectional area in sq. ft. Choose duct paths that minimize length and bends, but these are not prohibited. MUA is an entire topic by itself. The goal is to allow air to get from the outside to the kitchen, heated or not depending on circumstances, and lightly filtered, such that the pressure loss from outside to inside is low. In some cases, this may require a blower in the MUA duct. Many simple configurations are possible, along with more complex ones. Search out MUA threads here. Overall, the hood and MUA systems can reach the same price as a high-end range price, and easily exceed the price of an economical induction cooktop.