How to make a full adder using 1/4 DEMUXs and 1 NOR gate?
A dumb, but a universal approach would be to use the demux to implement a universal gate such as NOR. Each demux can implement 4 different functions of two variables on the following way:

     |^^^^^^|
     |      |--- A'B'
1 ---|      |--- A'B
     |      |--- AB'
     |      |--- AB
     |______|
       |  |
       A  B


Notice A'B'. It is the same as (A+B)', which is... NOR! 
So this proves that the task is possible and doable (or a stronger claim - that any logical function can be implemented using 1-4 demuxes).   

As a bonus you can get an AND gate and two components of XOR out of it, which will significantly reduce the number of such a a units in the final circuit. 

Note
If you are not allowed to use constants such as 1, you can utilize the provided single NOR gate connected to the two of the demux outputs as following: 

     |^^^^^^|
     |      |---______
A ---|      |---\ NOR \____
     |      |---/_____/
     |      |---
     |______|
       |  |
       A  A


Since these two outputs will be low for any A, the NOR will output 1 always.