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.

Any range hood extractor fan would be suitable. It depends on the aesthetics of your kitchen, and your budget as to which style would suit the best. Have you thought about a down draft extractor fan?

For the small amount of range hood that would be in contact with the undulating subway tiles, I do not see a problem.

it does look great but it could be cheaper and definitely much easier to employ a kitchen specialist company that can coordinate this job as you will need a cabinetmaker, a carpenter, an electrician and possibly a roof plumber to install a range hood like this.. in an ideal situation if you can organise all the tradespeople together and there's no existing cabinet to remove (and walls to repair that will also need a plasterer and painter) i would expect the job to take a full day with the cabinet built off site and quickly installed but it depends on the specific needs of the location eg is there easy access to the roof space and no restriction with ceiling framing to install the exhaust flue (or can the flue be directed out the wall but how does that affect the brick walls) and where is the power and can it be redirected inside the new cabinet and??? at least obtain a quote from a kitchen company to establish a base cost but with all the different trades for several hours each plus materials plus the cost of the range hood i would expect to spend at least $3000 or more without any surprises...it's not the cabinet that costs the money it's the complex cooperation of trades and unexpected labour time that's unknown and i would not attempt to start without coordinating the whole team needed to do the whole job