Suppose we have N systems prepared, and the first is in thermal equilibrium with the second, the second in thermal equilibrium with the third, etc., until system N - 1 is in thermal equilibrium with the N th system. Is the first system in thermal equilibrium with the N th ?
By repeated application of the Zeroth Law, we can state that all N systems are in thermal equilibrium with each other.
Does the Third Law insist that the entropy go to zero as the temperature approaches zero?
No, the Third Law insists that the entropy go to a constant value as the temperature approaches zero.
Is this statement true? Why or why not?: As the temperature approaches 0 o C , the entropy approaches a constant value.
Remember that 0 o C is equivalent to 273.16K and is therefore very distinct from absolute zero. The Third Law says nothing about what happens at 0 o C .
Is this statement true? Why or why not: every system will increase in entropy as time passes.
The statement is false for two reasons. A perfectly closed and isolated system at equilibrium will not exhibit any change in entropy. Secondly, the statement did not insist that the system is closed and isolated. When you place an ice cube tray full of water into your freezer, you expect the water to become more ordered and turn to ice. The system that consists of a well in the tray shows a decrease in entropy, but if we take the system large enough, say your entire house and isolate it from the environment, then that system increases in entropy as you freeze the ice.
In the jargon of the Second Law, what does it mean for a process to be path independent?
One familiar with multivariable calculus can recall that path independence has to do with the irrelevance of the specifics of a process but mere reliance on the initial and final state. In this case, the variables of a system can be altered in any way, and provided that the end state is identical to that obtained by some other way of altering the variables, the end result of the quantity in question, namely Heat, will be the same.