I have some questions about the voltage analysis (see image above).

* In this case, would the expression for Vp be Vp = V1-V2? More exact: is it the polarity of Vp that decides how the expression of Vp looks?
* Is the expression of Vp independent from the reference node, as long as the supernode around Vp is not including the reference node?

 

* In this case, would the expression for Vp be Vp = V1-V2? More exact: is it the polarity of Vp that decides how the expression of Vp looks?

Your expression looks good for me. If you reverse Vp voltage polarity then Vp = V2 - V1 or -Vp = V1 - V2
But for this circuit we already know all the voltage.
V1 = Vs and V2 = Vs - Vp so we don't need Vp = V1 - V2

* Is the expression of Vp independent from the reference node, as long as the supernode around Vp is not including the reference node?

Yes.