Here’s a suggestion: http://www.suncoastsheetmetal.com.au/cms/index.php/domestic/kitchens.html

there are several range hoods eg Miele that can be vented horizontally and even some that can vent downward under the floor and another that has the motor outside the walls but you'll need to check with manufacturers to clarify which one would suit as you need to know exactly where the wall and floor framing is located to make sure there's enough space for installation ..rear venting rangehoods don't use less cabinet space they just avoid having a pipe through the ceiling and roof...

Yes, but are you using the gap where rangehood was to centre, or the centre of the cook top. If you have cabinetry along the wall then you can hide the fact that the extractor fan/rangehood is not centre of the cook top. Or that is how it appears in your photos.

If there isn't a duct leading somewhere it's not ducted. Venting into the cavity doesn't sound right? Perhaps it was permissible when originally built? Our first house had a simple exhaust fan, of the type also used in bathrooms, above the stove and it went straight into the ceiling. That wasn't unusual in cheaper homes in those days. But these days kitchen exhausts should be recirculating or ducted externally to avoid build-up of potentially dangerous residues. In your situation recirculating sounds the most straightforward option. Alternatively you might be able to run the ducting out a side wall but the longer the ducting the less efficiently it works. Is it possible your rangehood has been installed incorrectly? We installed a rangehood in the 90's which could be configured as recirculating or as ducted and It took a few attempts to get the thing working properly.