I can't answer every part of that question, but I can tell you what I do know. In the United States, weather generally moves west to east. If I had to guess, it would probably be because of the movement of the jet stream. That would probably carry systems from the west to the east. Europe would probably see something similar, but not the same. I can't say for sure what happens over there.
With hurricanes, storm systems buck the trend and move mostly from east to west. This is because of easterly winds, which blow westward. The storms are pushed by these winds to the west. The usual turn to the Northeast is thanks to the Coriolis effect, which is a deflection of an object on a rotating plane. This deflects hurricanes to the "right," relatively speaking. And yes, these storms can and do affect Europe. The remnants of Hurricane Maria in 2005 caused significant rainfall in Norway that killed three. Hurricane Vince, in the same year, impacted Spain and Portugal.
In the Southern Hemisphere, I would assume that everything is switched over. Weather systems move from east to west in the land Down Under. Hurricanes (or Willis Willis, as they are called by Australians. The country's weather service calls them cyclones.) down there move from west to east, carried by the westerlies, and deflect to the "left," turning back to the west and following regular patterns.
This is just a guess. Patrick, am I right on this?