Where did Nano get money from? For example, for shopping.   

The anime doesn't show that Prof. or Nano works. 
Does Prof. and/or Nano have a stable income?
The money seems to be handled by Nano, so I doubt Hakase sells her inventions herself. (otherwise she would just buy loads of chocolate sharks)

It is presumably from a relative, or the money comes from a patent on an invention (with the money going to Nano).

No relatives are ever mentioned, so it's probably some form of research funding. (I personally think one of Hakase's inventions actually backfired at some point, turning her into a child, that would explain why she gets it and that Nano handles it.)