Paging /u/Domini_canes on this one.

The Vatican was indeed neutral during WWII, adhering to its treaty obligations from the Vatican Accords signed in 1929. There is some controversy regarding Pius XII's actions regarding the Holocaust, which I discuss here in some depth. Another recent thread asked specifically about the book Hitler's Pope, and I discuss the many problems of that thesis in that thread.

How severely was Vatican city damaged

Damage was very slight relative to the scale of the war. The Vatican itself was bombed twice: November of 1943 and March of 1944, both times were during the German occupation of Rome. The first caused no casualties and a bit of damage in the Vatican gardens. Just who dropped the bombs was disputed at the time and isn't entirely clear today. At first it was blamed on the Germans, Americans, or British. A recent book claims it was done by Italian fascists, while another claims it was the British.

The second bombing killed one and wounded another, damaging the Palace of the Holy Office and the Pontifical Urban University on Janiculum Hill (which is outside the walls of the Vatican itself but is part of the territory controlled by the Vatican according to the aforementioned Lateran Accords. If I understand correctly, it is in the area in dark black on this map, just south of the Vatican proper). This attack was made by an Italian plane.

were they considering any kind of war goal or objective?

The Vatican had no military force (only a force of guards and police, under 500 total if I recall correctly) and had only diplomacy as a way to attempt to influence events. There were a multitude of diplomatic initiatives with varying levels of success. The Vatican's main goal was to mitigate the suffering, particularly for Catholics and noncombatants. In this they had only modest success. Attempts to broker peace deals beginning before the war began and during its duration were rebuffed. Attempts to get nations to stop targeting noncombatants were ignored or rejected, such as the many appeals Pius XII made to FDR regarding the Allied bombing campaign.

Did they have contingency plans based on how the war was going?

We know only a little about this, as the archives for Pius XII's pontificate are not yet open for researchers. What has been released focuses on his response to the Holocaust rather than his reasoning regarding various contingencies. He certainly wished for an Allied victory (this is clear from diplomatic dispatches already released by various governments), but was leery of the Soviet Union to say the least.

I hope that answers most of your question. Followup questions from OP and others are always encouraged.