assuming that sea water has a density of 1.05 g/cm^3, what is the density of a submarine running submerged?

wouldn't you need to know some other information; like the mass or volume of the submarine?

No. The average density of the submarine would have to be the same as the surrounding fluid, if it is neither rising nor sinking.

Think of the Archimedes Law and you should see why this is so.

Yes, you are right. In order to calculate the density of a submerged submarine, we would need additional information such as the mass or volume of the submarine. Density is determined by the mass divided by the volume, so without knowing either of those values, it is not possible to calculate the density accurately.