"Inverse problem" for resistor network: finding applied voltages to produce desired currents
You have a system of 5 equations such as: 

$$i_1 = \frac1R(V_1-V_2)$$

You can also determine the voltages that you cannot control from nodal analysis in terms of only the voltages that you can control and the resistances. Upon substitution of these equations into the system, you will have only three independent variables.

From here, use your favorite linear system solver (or do it by hand) to see if 1) a solution exists and 2) what such a solution is.

For question 3: you will want to use your favorite optimization package (or by hand) and define your objective function as the error (you mentioned least squares). Then solve the optimization problem for minimum error.