finding missing current and voltage in a circuit involving superposition
from cases 1 and 2, I should have I1 = 4 + 5 = 9 mA, but it's 8mA.


4 mA + 5 mA is what you would get if you applied 2 V on the left and -10 mA on the right.

But that isn't what you were asked. You were asked what happens if you apply 4 V on the left and 8 mA on the right.

So, to use superposition, first use test result 1 to find out what happens if you apply 4 V on the left (with the right port open). Then use test result 2 to find out what happens if you apply 8 mA on the left. Then sum those results to get your final answer.


  Hence, I thought that the black box consumes 1mA;


The black box contains only linear elements. Therefore it can't "consume" any current.

Edit

OK, let me spell things out.

When they say the system is linear, it means you can write these equations for the unknown variables

$$I_1 = G_{11} V_1 + A_{12} I_2$$
$$V_2 = A_{21} V_1 + R_{22} I_2$$

where \$G_{11}\$, \$A_{12}\$, \$A_{21}\$ and \$R_{22}\$ are characteristic parameters of the black box that you need to discover. The names of the parameters are arbitrary, but you should think about why I might have named them what I did.

Your two test measurements give you enough information to determine the unknown parameters. 

Then you can plug in any \$V_1\$ and \$I_2\$ you like (for example, the ones you were asked to solve for), and find the resulting \$I_1\$ and \$V_2\$ (for example, the result you were asked to find).