Does dinitratomethane exist?
After some more research I'm pretty much convinced in its existence. As Mithoron noticed in the comments, the name dinitratomethane is somewhat confusing, and is indeed not preferred.

I suppose the most accurate IUPAC-compatible one would be (nitrooxy)methyl nitrate. It also turned out that in the literature (mostly on regulated and hazardous chemicals) this compound is generally known as methylene glycol dinitrate.



The second volume of a classic handbook "Chemistry and Technology of Explosives" [1, p. 155] gives the most extensive description I discovered (reference numbers are updated):


  Methylene glycol dinitrate, or "nitromethylene glycol", is an oily liquid having a boiling point of $\pu{75-77 ^\circ C}$ at $\pu{20 mm Hg}$.
  
  It is a strong explosive, capable of dissolving collodion cotton. However it is not stable, being readily hydrolysed and therefore it is of no practical significance. 
  
  A method of obtaining this explosive, by the nitration of moist 
  paraformaldehyde was patented by Travagli and Torboli [2]. They utilised the well-known property of formaldehyde of reacting like methylene glycol in the presence of water. 
  
  According to Travagli [3,4] methylene glycol dinitrate is obtained as follows: 
  paraformaldehyde is dissolved in sulphuric acid and added drop-wise at $\pu{3-5 ^\circ C}$ into a mixture of nitric and sulphuric acid. The resulting oil is separated and washed with water to give a final yield of $35\%$.


The recent handbook of liquid explosives from 2015 [5, p. 227] contains virtually the same information without any updates, which indicates there simply hasn't been much of an interest in this compound due to its instability and lack of possible applications.

References


Tadeusz Urbanski. Chemistry and Technology of Explosives, 1st ed., Pergamon Press GmbH, 1965; Vol. 2.  
G. O. Travagli and Torboli, Ital. Pat. 338080 1935.   
G. O. Travagli, Gazz. Chim. Ital. 1938, 68, 718. 
G. O. Travagli, Chimica ed Ind. 1938, 20, 389.  
Jiping, L. Liquid explosives; Springer Berlin Heidelberg: New York, NY, 2015. ISBN 978-3-662-45847-1.