weather influence on some moves

Judging by the first screenshot, we're looking at a 20% increase in power, rounded up to the next whole number.

Will there be like a Niantic medium who will state that Europe is currently giving off good energy so psychic mons get an increase??

Wait, this Charizard on photo have Fire Spin and Overheat?

I think it's a great addition. Add another layer to the game.

Might give us that extra push we need to finally solo Flareon or Jolteon

I think it would be cool if sunny weather reduced the cooldown of solar beam by 20% (from ~4.9s -> 4s)

I live in (usually) sunny California. This may put a good Moltres above Dragonites as DPS generalists.

How are they going to know the local conditions everywhere in the world simultaneously at all times?

This can not be accurate - they don't have the capability to make it accurate.