This is the hood installed

and this is after removing the hood. The backsplash (which we’re planning to replace/update) starts 11” below the cabinet so the 7-10” height hood will leave a gap but I’m thinking I can patch the Sheetrock for the short term and fix it up for the long term once we redo the backsplash/counters

Because cooking plumes spread out as they rise, if the hood is not large enough some cooking effluent will not be captured. Whether a small increase if height above the cooktop will cause much escape cannot be determined without knowing the dimensions of the hood entry aperture and the dimensions of the cooktop. In general, however, hoods that overlap at 30 inches probably will be OK at 36 inches gap.

You can mount the new thin hood lower and put a trim piece in the gap.

What is that duct's sectional area or size, and where does it go.