Installing PEX pipes close to heating vents
Great questions! Yeah, I agree that the 1/4" AIN'T RIGHT. Have him replace those sections. A body-spray setup needs all it can get.

Being close to Ductwork is fine, the hot water is hotter than the ductwork will ever get. FYI, the Blue PEX is just coloring and is exactly the same as the Red PEX. However, PEX cannot withstand a furnace flue's/exhaust pipe's temperatures, ensure PEX is not touching nor even within 2' (my requirement) of that Ductwork.

Yes, there is a way to test the Plumber's crimping workmanship. It's a Go/No-Go gauge and he should be using one himself...likely molded/carved onto his crimper. See https://www.menards.com/main/plumbing/plumbing-tools/pex-tools/nibco-reg-go-no-go-pex-crimp-gauge/px01382r1/p-1444449294533.htm

So many fittings? Yep, kind of kills the "PEX advantage", doesn't it...soldering ain't slow and only gets A LOT faster in your kind of shower setup. I'm not a fan of PEX. It's only good for 50-years and copper is good for OVER 100-years (in most areas)...which house would you buy...sell in 20-years or less.

By the way, while you have some access to duct joints, liberally wrap the joints with Aluminum Foil Tape...the real duct tape.