If a product exists for a set of objects in a category, then is there a product in the category for every subset of those objects?
Not necessarily.
For example, in a poset, viewed as a category, the product of a family of objects is their greatest lower bound. Now it's quite possible to have a poset $P$ and sets $X\subseteq Y\subseteq P$ such that $Y$ has a greatest lower bound but $X$ does not.
The simplest example has $|X|=2$, $|Y|=3$, and $|P|=4$. Take $P=\{a,b,c,d\}$, with $a,b\leq c,d$. The greatest lower bound of $Y=\{a,c,d\}$ is $a$, but $X=\{c,d\}$ has no greatest lower bound, since the two lower bounds for $X$ are $a$ and $b$, which are incomparable.