Question

why do nitro compounds dissolve in sodium hydroxide?

Answer 1

The compounds containing nitro groups when dissolved in basic
medium, the following things can happen.
The base can take the α-Hα-H of the nitro-group, which will create
a negative charge on that α-Cα-C, which can further delocalise
with the Nitro-group (similar to Nitro-Acinitro tautomerism). 
So, the whole compound will be in the state of a Resonance
Hybrid where there will be partial double bond character
between α-C,Nα-C,N and two OO atoms. and also there will be
a δδ- charge on the α-Cα-Catom as well as on the OO atoms. ,
and δδ+ charge on the NN atoms.Due to such salt like compound
formation, it becomes soluble.
This δδ- charge created on the Carbon atom is stable if the side
chain is short, but starts to get destabilised as the side chain
grows longer. So, I think the solubilities of the Nitro-compounds
decrease as the side chain grows longer, and the compound
with 10−1210−12 C atoms may not be at all soluble in base
because of such large Hydrophobic part and destabilisation of
the δδ- charge.