Edge + Zwift + Smart Trainer functionality

Yes it is possible.  Pair the trainer to your edge as a power meter, not as a trainer.  This will let Zwift control the trainer and your Garmin will just record/show the power readings.

You can also pair it as trainer. As long as you don't do a workout your Edge won't control your Core.
The other way round is also useful if you want to take part in a Zwift event but also want to do your activity. E.g. you want to ride Tour de Zwift events but follow your training plan. In Zwift you can either be on an event or perform a workout. But it works when your head unit (Edge) controls your trainer and Zwift just take the power reading. Load your workout to your Edge and unset your Core from Zwift's controllable source.

As stated by others, yes it's possible.

Be aware though, it's another reason for Zwift to mess up connections.   We all know Zwift suffers from issues at the best of times, and in our cycling club, running Garmin and Zwift together is the biggest cause of problems.   It doesn't impact everyone, but it does many.

Give it a go...it may work for you, but be aware that is you start getting drop outs then this is likely to be the reason why.

Thank you very much for the answers. This is really helpfull.

Can you tell what will happen if I don't unset the Core in Zwift - will it then be controlled by both units?

kasperly Are you saying your Trainer is paired to your Edge and to whatever device is controlling Zwift? If so, it should be whichever device you start the ride or workout with. 

If you start a ride on Zwift, you can technically just use the trainer as a power meter and record an activity on your Edge without controlling the trainer.