LM78xx Reverse Polarity and Reverse Bias
Yes, you can do what you propose.

I would recommend larger output capacitors in parallel with the 100 nF they have now. The regulator cannot react instantaneously to step increases in load current so the capacitors are there to provide the current until it does. Capacitors with a low ESR are accordingly good for this, try ceramic parts. Details of how to calculate the values can be readily found on the internet. From experience (gulp), I would use the following. On the 5 V rail, put a 10 uF and a 100 nF close to the regulator, put a 10 uF and at least one 100 nF very close to the load. On the 12 V rail, put a 10 uF and a 100 nF close to the regulator and a 100 nF close to the load.

Incidentally, the reverse diode across the regulator is used to discharge the regulator's output decoupling capacitor when the supply is removed and the regulator has a light/no load. If the regulator's load will discharge the output capacitor anyway then you don't need that diode.