Even if that ceiling vent incorporates a fan and exhausts to the outside, it does not qualify as cooking ventilation. It might be able to clear the air after some period of time in which grease particulates and water vapor have settled onto surfaces.

Cooking ventilation captures and contains cooking effluent plumes rising and spreading from pans on the cooktop and ejects them outside. (Recirculating systems with adequate filtering may also qualify.) This keeps grease off of household surfaces and provides various health benefits, including ventilation of gas combustion products from the cooktop burners.

Cooking ventilation comprises, at a minimum, a hood large enough to overlap the cooking effluent plumes, a grease filter/fire-stop such as a baffle array, a blower (in the hood, in the duct, or externally mounted), and connecting ducting. There are minimalist hoods with blowers that are better than nothing, but won't adequately address wok cooking or meat searing, and larger systems of increasing complexity that can do so. The vast number of hood and make-up air (MUA) threads in this forum should be sampled to provide insight into the requirements and complexities of cooktop ventilation.

Over the range (OTR) microwave ovens cannot provide the plume coverage or the air flow requirements of cooking ventilation.

The question about "exterior wall" asks which wall in the image, if any, is in contact with the world external to the house. If the wall behind the range is an exterior wall, then hood system ducting is greatly simplified, unless there are issues with venting into that particular external area.