You may find an earlier post of mine regarding Pius XII during WWII to be interesting.
The Vatican was bound to strict neutrality according to the Lateran Accords, Article 24 to be specific. As such, choosing sides was not possible.
The Vatican itself was hit by bombs twice during the war: once by the (edited) Axis and once by the Allies. Neither produced casualties and damage was light. Other than those two incidents, Vatican property was not violated. There are stories surrounding a potential plot to kidnap the pope and remove him to Germany, I am among the historians that doubt the veracity of these claims. I haven't read the most recent scholarship on the issue, but I have yet to see any substantiation beyond memoirs that make the author the hero of the story.
I would be happy to answer any followup questions you or others might have. I didn't want to clutter up this post with a massive cut and paste from my earlier answer, as it hit the character limit for posts on reddit a number of times.