Q1. Can a unit trace supply through a neutral country?
Q2. This question has to do with the attrition phase. If a unit blocks the supply of an enemy unit as the last action of the round is the enemy unit eliminated during the attrition phase or do they get marked as out of supply and become eliminated during the next attrition phase?
1. If you're using "neutral" in its usual meaning (a country that is not yet at war), then no, your units can't trace supply through it.
The reason I have to ask what you mean is that early versions of the rules, and the Consimworld FAQ, also describe countries with no units (Albania and Persia) as neutral, even though Albania is at war (and may be entered by both sides) at the beginning of the game. Albania (and Persia after Turkey enters the war) are not neutral countries in the usual meaning; they are Allied countries with no units of their own.
Oddly enough, there is also a special rule, [11.1.12], which allows empty Allied controlled spaces (but NOT units!) in Albania to trace supply over the "IT only" dotted line to Taranto even while Italy has not yet entered the war.
2. Being out-of-supply is immediate (and attrition is resolved all at once, so if two units are sitting on each other's only supply lines during an Attrition Phase, they both die). This is something the Central Powers, especially, have to watch, because the Allies always have the last action in each turn.