CMOS input impedance
I would guess that the "problem" is mains pickup, the CMOS counter IC has an incredibly high input impedance and so is very sensitive to noise if left 'floating' ie no connection from an output.
Impedance is just a fancy word for resistance but allows for weird AC effects like capacitors and inductors whose 'resistance' is different at different frequencies. We generally refer to DC circuits as having resistance and AC circuits as having impedance.
Try connecting a resistor from the input to 0v (your negative power rail), anything would do, 1Meg, 220k, 10k even as low as 1k ! this will reduce that input impedance to something more resilient (and less static sensitive !).
As for the mains power supply, I should think the capacitors are correct and won't be better swapped to 1uF, bigger values are actually worse at high frequencies (produced by the 7805 regulator IC). If you want to add more capacitors then -add- them, a 1uF, 10uF, 220uF would be fine added but leave the little one there as well !