I would actually think that cubane incorporating nitrogen atoms into its structure would be synthetically easier,
especially if the structure was symetric, as in the case shown in the picture of "tetraazacubane". Coupling nitrogen to carbon, rather than forming a carbon-carbon bond, is usually far more simple. However, with each carbon atom bonded to three separate nitrogen atoms, the cage may likely be chemically unstable since nitrogen prefers to triple bond to carbon. I also tend to think the cage would spontaneosly come apart because the carbon-nitrogen bonds in such an an arrangement (similar to hexamine) would have a small equilibrium with any traces of water that would inevitably exist, or any other solvent that is capable of ionizing. Similar to the other topic in this forum about geminal amines, molecules containing a carbon atom that is single bonded to three different nitrogen atom (no double bonds) are almost nonexistant.
As for its stability after adding nitro groups, I would remind you that all attempts to prepare 2,4,6-trinitro-1,3,5-triazine have been unsuccessful. Adding nitro groups to tetraazacubane would have a severe complication, in that the acidity of the nitrating agent would almost certainly pull apart the carbon-nitrogen bonds.
It can be fun to imagine a potential new molecular structure, but this is very problematic for energetic compounds. Because of all the nitrogen-carbon bonds, often it is not possible for such theoretic molecules to exist. My recommendation would be to consider all possible resonance states and ways that the molecule could ionize, and any equilibrium that the molecule would have with water, to first see whether the molecule would come apart.
If you are connecting three nitrogen atoms to a carbon, one of the nitrogen atoms should be double bonded to the carbon. Also be aware that nitro groups will not stay on a carbon that also has an amine group on it (unless the amine is fully substituted, without any hydrogen atoms). When a methylene group (CH2) is connected to two other nitrogen or oxygen atoms, there is inevitably equilibrium with water, which could cause your molecule to hydrolyze apart if the equilibrium is unfavorable. One example, attempts at nitration of melamine to a form a trinitramine
(C3N3)(NHNO2)3 have not been successful.