Can you think of any entire function that has zeros at regular intervals (consider periodic functions)? Simple transformations of one of those should provide a counterexample to both (b) and (d)--in particular, let $f$ be one such (appropriately transformed) function, and show that neither $f$ nor $g$ is constant.
Note: By definition of $g$, we have $f(\frac1n)=f(\frac1n+1)$ for all positive integers $n$ if and only if $g(\frac1n)=0$ for all positive integers $n.$
Now, an entire function is defined and analytic on an open connected set--namely, all of $\Bbb C$--so by the Identity Theorem, if two entire functions agree on a set $S$ that has some accumulation point in $\Bbb C$, then they are in fact the same function. Observing that any constant function (and in particular, the constant $0$ function) is entire, and that the set of reciprocals of positive integers has an accumulation point at $0$, what can you conclude about an entire function that has a zero at $\frac1n$ for every positive integer $n$? This should let you conclude that (a) is true, and (along with the Note) that (c) is true.