This was part of the OP. Just to clarify, if you look at 250.118 both the MC and EMT should be properly terminated. So, if you properly terminate the EMT into the 4" metal box with a fitting then the MC must also be terminated with a fitting when it enters the EMT. That way the EMT will be mechanically connected to the MC.
I agree.
If you choose to sleeve the MC through the EMT (just to make it a straight and a neat run) and terminate the MC to the box with an MC connector you will still need a "from-to" connector (like the one I posted) where the MC enters the EMT to mechanically connect both together. It does not have to be bonded to the EGC (in case anyone interpreted my original post that way).
I disagree with that part. In this instance the piece of EMT is not used as a raceway, it doesn't even need to be listed EMT or other metallic raceway - it can be any metal tube or pipe - it is nothing more then a chase to route the cables through. It can be considered as much a part of the building as a wall stud or floor joist, could be small enough for only one cable to fit through or large enough for several cables to fit through. Ampacity adjustments may apply - but are stricter for NM cable in this instance then they are for MC cable.
After reading your second part again, maybe I am not seeing what you are trying to describe. If you have a single cable and want to use a cable to raceway adapter -then run "stripped" cable through a raceway to a box - you need proper raceway termination at the box.
What I was talking about was a sleeve where the cable or even multiple cables enter one end of the sleeve with no connector or fitting pass through the sleeve and exit with no connector or fitting. The purpose of the sleeve was either physical protection or dressing up the cable(s) as they passed through a particular space where they are exposed. The cables then continue on to a box, cabinet, etc from both ends of the sleeve, which could be a few inches or a hundred feet away.
Since this sleeve is just a "mechanical chase" it could even have other items in it like hydraulic or pneumatic lines, but one may need to watch things like steam lines or you could have temperature issues withe the cables.