You mean like the vent on the left? Sure.

Yes, you might need to order your insert/hood with the hole on the back instead of on top. I did the vent out the back in my previous house. On the outside you will have a "box" with a one-way damper and wire screen to stop critters from getting in.

Vertically we would have an issue because there's a bedroom right above the kitchen. So we do not see any way of running the vent to the roof..

Direct vent would require us to run the pipe up and out the side of the wall of the house which happens to be brick.. however I guess it can be done.

Make sure you or your installer follows code, in the U.S. there are minimum distances an exhaust termination must be from other building openings, like windows, HVAC intakes, even roof eave soffit vents.

Direct vent would require us to run the pipe up and out the side of the wall of the house which happens to be brick.. however I guess it can be done.

The vents of our condo building's lower units are on the exterior brick walls. (Upper-unit vents are on the roof.)