Reduction from Partition problem to 3-Partition problem
To answer your bolded question: The sum of the numbers in $Y$ is $\frac{3m}2$. Thus $A$, $B$, and $C$ each need to be of size $\frac m2$. This can only be achieved by partioning as in the first direction.

Regarding your second question: You can not be sure that $\frac m2$ is an integer and you have to pay attention to it, in order to get a formally correct reduction. 

The easiest way to address this is by having your reduction function distinguish two cases: If $\frac m2$ is an integer, it proceeds as you describe. If $\frac m2$ is not an integer, it exploits the fact that in this case the answer to the original question ("Can $X$ be partitioned?") is easy. I'll leave it as an exercise to figure out the details of this case.