Sorry didn't see this been very very very busy.
Snow - starts off in clouds as snow. Then depends on what it travels through.
If it goes through warm air it melts! but.... as it melts, if you get dry air in that layer of air you can get something called evaporative cooling. This cools the air down espcially if you get heavy precipitation and it can make it cold enough for it to remain as snow.
Now, if you do have heavy snow, as long as the air is cold above it can last even when the temp in the air close to ground level is above zero.
snow definitely is partially dependent on humidity, low humidity=more likely to be snow.
direction only matters in that the atlantic is relatively warm so showers that are feeding in from the west tend to be from warmer air and generally more likely to be rain. Of course if that warm rainy air hits cold air from the east them where that happens = much snow.
hail - can start off as snow, or rain. If it is snow then obv it has melted then refrozen.
sleet is just when it can't make up its mind!