An apperent contradiction to 0's of holomorphic function are isolated! I have a few questions about functions of this form:
$\displaystyle f(z) =\sum _{n=0}^{\infty} \frac{a_n}{Z-b_n} $


*

*If I put $a_n = 2^{-n}$, $b_0 = 0$, and $b_n =1/n$  for $n>0$, then we get a well-defined  holomorphic function $f$ on  $D = \Bbb C - \{ U \{1/n) \cup \{0\}\}$. It has simple poles at $(1/n)$.
Technically $0$ is not a singularity of $f$ (as singularity means function is defined in a deleted neighbourhood of it).  What kind of behavior does $f$ have at $0$ ?    If it is similar to a pole (i.e. $ |f(z)|\to \infty$ as $z\to 0$ inside $D$), the $\frac 1f (0) = 0$ which is not isolated  as $\frac 1f (1/n) = 0$ for $n>0$ which is a contradiction. (The zeroes of holomorphic functions are isolated)
What are zeros of $f$ ? One can easily conclude that any zeroes of $f$, if they exist, are positive reals.

*Now put $a_n = 2^{-n}$ ,  $b_n  = \{ 0,-1,1,-2,2,...\}$. Is $\sin (\pi Z)  *f(z)  $ an entire function ? If it is, can we generalize this to any function g having simple 0's at arbitrary $b_n$?
 A: 
Technically $0$ is not a singularity of $f$ (as singularity means function is defined in a deleted neighbourhood of it)

What you said in parentheses describes an isolated singularity of $f$. The function you describe in (1) is not holomorphic at $z=0$, and does not have an isolated singularity there. 
Whether or not to consider $z=0$ an essential singularity is again a matter of definitions. I think this term should be used only for isolated singularities. For one thing, this allows for a concise statement of Picard's theorem: in a neighborhood of essential singularity the function attains all values except at most one. 
You could invent a special term for points such as $z=0$ in this example, but what would be the point? (No pun intended.) There is no theorem that I know of that says: if $f$ is not holomorphic at $z_0$, then ... The terminology for various types of isolated singularities is introduced not because they are "bad" points, but because they are actually not so bad: there is a Laurent series around, and we can look at how many negative powers it has.

What kind of behavior does $f$ have at $0$? If it is similar to a pole

Weird one. Not similar to a pole at all. 

What are zeros of $f$ ?

They depends on $a_n$ and $b_n$ in a mostly inscrutable way. Even when the sum is finite (all but finitely many $a_n$ are zero), the question amounts to finding the critical points of a polynomial, knowing its zeros. There's no formula for that. 
A: Regarding your second question, it's true that if $f, g$ are holomorphic in some pointed neighbourhood of $x_0$, and if $f$ has a simple pole at $x_0$, and $g$ extends to a holomorphic function that has a simple zero at $x_0$, then $fg$ extends to a function that is holomorphic at $x_0$. More generally, if $f$ has a generalized zero of (possibly negative) order $k$ at $x_0$, and $g$ has a generalized zero of (possibly negative) order $n$ at $x_0$, then $fg$ has a generalized zero of an order $n+k$.
