How far does reforming a capacitor help? Does it have any adverse effect on using such a capacitor on components?
I don’t know how the Eico does the reforming, there are variations on the process.  The typical way is to use a power supply set to the rated voltage of the capacitor, plus a resistor large enough to give a reforming current of some “reasonable” value.  Some people recommend about the same as the normal leakage current of the capacitor, others recommend something much higher.  

Monitor the voltage across the capacitor during the reforming process.  It will probably start low and rise over time.  When it is at or near the rated voltage the process is done.  If the voltage never gets near the rated voltage the capacitor is hopeless and don’t use it.  

Edit:  Reforming seems to be mostly for older capacitors.  I have read that modern electrolytes make leakage over time less of a problem.