Can I put graetz circuits' outputs in series and parallel?
That's fine with certain constraints:


The AC voltages have to be isolated from each other.
The sum of the AC voltages on the left has to match those on the right.
Add the capacitance between OUT- and OUT+.


The risk is that if the output voltage of one side is a little higher than the other then it will supply most of the current and may overheat.


  And can I do anything to satisfy the second constraint? The individual AC power will come from separate coils and I assume they will never produce equal power, since the whole device is quite crudely crafted.


Parallel connection of transformer secondaries wound on the same bobbin even has this problem. It is resolved by the internal impedance of the coil. As the coil current is increased the voltage droops a little due to the internal resistance. This tends to be enough to balance out the currents.
In your application you could add a series low-value resistor on each leg. You could check the effect by measuring the voltage drop across each series resistor to calculate the current from each leg.