Side question: what is the minimum laser power required so beam can be seen at night?
It depends on the frequency of the light.
Since here, we're discussing 532nm greens, even 5mW can be seen in the sky at night.
However, the ANGLE at which you're viewing the beam will have a great effect. Under city or suburban light pollution, or bright moonlight, or fireworks, like in the OP's example, it takes probably 30mW or more to be seen from the side when standing more than 20-30 feet away from the laser.
My experience is much like InfinitusEquitas just above my post here. I have some 30mW greens I was paranoid about sky/star pointing with, until I had a friend point out a star, and I walked up my driveway to watch him from 50 feet away, and I couldn't find the beam at all.
Holding the laser and looking right up the beam, or standing right next to the person and looking up the beam, it's much brighter.
Pitch black countryside, only lit by starlight, it's much easier to see the beam from the side from a greater distance. Even then though, it will require fog, smoke, or a few 100mW's to be really visible from the side at any great distance, like 100's of feet.