1.
Yes, it is possible that when mains power comes back on, it is interrupted by several rapid 'switch bounces'. These might carry spikes/ hash/ dropouts through an attached DC power supply.
A power line conditioner could go a long way toward helping to solve your problem. Perhaps a surge protector.
(Note: The varistor, or surge absorber, is only able to absorb so many spikes at a certain energy level, and then its useful life is over.)
2.
Static charge is a hazard if your system is exposed to the elements or located high above ground. It may or may not contribute to your problem.
Anyway a static charge arrestor is typically recommended. (The same kind as for TV aerials.)
3.
Some pins on the microcontroller may need to be tied to a supply rail, particularly input pins. If they are left floating, they can adopt unpredictable volt levels.
Guidelines state that unused input pins should be connected to the power supply or to ground.
4.
The reset switch may need attention, either regarding item #3 above, or in regard to keeping it activated until the power supply settles.
(I have no direct experience with microcontrollers, but I did install a reset switch in my Commodore computers.)
There have been previous threads asking about a delay circuit at the reset pin, in order to ensure that a microcontroller will wait for a moment after power-up, before it starts running.
Simple method that will work in the case of a reset pin that needs to be pulled low to halt the microcontroller. The IC begins to operate when the reset pin goes high.
Component values are not critical. The high-ohm resistor is added so the capacitor can discharge during a blackout.