No, unfortunately not. All the technology has to fit into such a stick.

I just spoke with a supporter from AirTame who says they are in the final test phase to not only operate screen mirroring with the driver but also to expand a monitor so that you would then have dual screen...

Or is there something like a possibility to send the picture to the TV via network/Wifi or a virtual 2nd screen via network/Wifi. The receiver box should be of any size...?

I guess that's because there is no chip yet that contains all the components and that's why the Purelink probably has a circuit board in it that gives different chips/components together this function. If some chip manufacturer would now cram everything into one chip then that would be technically possible I guess, the question is only when will there be such a chip...