Alex's papa has dual citizenship himself actually. German and British.
Applicable via the Sophia Naturalisation Act 1705 and thanks to Ernst's father Ernest.
I don't know about Alex. I was mulling over this question recently actually. I came to the conclusion that she at least has Monégasque citizenship, which doesn't impede her right to either throne.
None of the Princely family needs citizenship or residency to reside in the countries they currently do (temporarily at least) because we have this lovely law in the EU where citizens of member states (Monaco isn't technically a member but it is treated as part of the borders of France, so it qualifies - Monaco is part of the VAT and customs areas of the EU and is part of Schengen) can travel to other member states and live/work there without a permit. Therefore, Caroline and Alex - technically French citizens under EU law - could live in France. As could Ernst.
So to surmise, I think Alex probably has just the one passport - a Monégasque one.
As for the Catholicism thing, it's perfectly fine for Alex to live in a Catholic country, with a Catholic family, going to a Catholic school and attending Catholic masses (Easter
) as long as she doesn't take communion in the Catholic church or "profess the Catholic faith". Besides, having been officially baptised in the Protestant Church she would, under the laws of the Holy See, have to convert. Children can convert (with permission and support of parents and a priest) but they would have to learn about the sacrament to partake in it. That requires sacramental and RCI courses and time. It's all just a bit too much hassle, to be honest, unless Alex
really wanted to become a Catholic. There's nothing stopping her from privately falling in line with the Catholic teachings yet still remaining Protestant. She'll just be a more conservative one.
(no insult meant to other Catholics of course, being one myself. Nor Protestants
)