Just to cover the basics - does the activity recording include also the heart rate data?

yes

So I double-checked (I’m using AW too) and my strength trainings do include calories; it should mean it's just a matter of configuration. Let's try a couple of things.

When you say “weight added” - make sure it's updated both in your Strava profile settings as well as in Apple Health. 

Also: what if you try recording a short testing activity via the AW Workout app (i.e. not directly via Strava), and sync it to Strava afterwards? Are the calories included then? (Note - not suggesting this as a solution, rather to rule out possible causes.)

And lastly - you say  you don't see calories for bike rides and strength trainings. Does that mean that other types of activities do include the calories information?