Alright, that sounds pretty good.
So, the communication between the Pebble and SkippingStones works.

That it is not doing anything when you receive a "real" e-mail is expected as this is not implemented yet.
Essentially, what is missing is that SkippingStones needs to get a notification about the new e-mail.
As far as I know, the preferred way to get that information should be via dbus.
The implementation of this usually is also pretty much straight forward.
However, one needs to know the correct dbus interface name, method/signal name etc. that is used for getting this information.
Of course, this all assumes that information about e-mails is available via dbus at all.

One starting point would be to use "dbus-monitor" in a shell on the Jolla and see what is going on on the system and/or session bus.
If anyone else has further insights into this, all information is highly appreciated.