What role does ring $R$ play in formal group? This is related to Iwasawa's Local Class Field Theory, Chpt 4, sec 1's Lemma 4.2 proof. 
Let $R$ be a commutative $k-$algebra s.t. $char(k)=0$. Let $G_a(X,Y)=X+Y$ be a commutative formal group over $R$. Then one can show that $End(G_a)\cong R$ by $R\to End(G_a)$ through $r\to rX$.(This is not hard to achieve. Consider $f:G_a\to G_a$ s.t. $f(x+y)=f(x)+f(y)$ with $f\in (X)R[[X]]$. Take derivative against $y$ and set $x=0$. One gets $f'(y)=f(x)=a\in R$. Matching degrees on both sides to deduce vanishing of higher order terms. Thus $f(X)=aX$ only.)

$\textbf{Q:}$ What role does this $R$ play here? Consider a finitely generated Abelian group $G$. Clearly $End_Z(G)$ is a ring. In particular, $G$ is $End_Z(G)$ module. $G_a$ is merely abstraction of the additive map $G\times G\to G$. This $G_a$ is really a pair, $(G_a,R)$ where $R$ is the coefficient ring. How should I interpret ring $R$? Or should I interpret $G_a$ as a family of groups $G$'s addition law s.t. each $G_a$ is equipped with its corresponding $End_Z(G)$ for each $G$ where $R$ becomes $End_Z(G)$?

 A: I think this is an oft-told tale, with many variants, as befits traditional story-telling, and I’m sure you can find out more pretty easily. Here I can give you only my own very partial account, and others will emphasize other aspects of the story.
The additive formal group law $\mathbf G_{\mathrm a}$ is just one of many formal group laws. While thinking of it, you might always keep in mind the other well-known group law over $\Bbb Z$, namely the multiplicative formal group law $\mathbf G_{\mathrm m}(X,Y)=X+Y+XY=(1+X)(1+Y)-1$. With the second formula, you see that if $n\in\Bbb Z$, there’s the $[n]$-endomorphism, amounting to combining $X$ with itself $n$ times (if $n\ge0$) using the group law. And explicitly, you get $[n](X)=(1+X)^n-1$, even when $n<0$.
Now, it’s a theorem that if your ring $R$ is a $\Bbb Q$-algebra, that is, when every $1/n\in R$, then every formal group law $G$ is isomorphic to $\mathbf G_{\mathrm a}$, and in particular the endomorphism ring will be isomorphic to $R$, by the map $\text{End}_R(G)\to R$, $f\mapsto f'(0)$.
But this is emphatically not true if $R$ is not so special; for instance, if $R$ is the integer-ring of a number field, and $G$ is the multiplicative formal group law, then $\text{End}_R(G)=\Bbb Z$, no bigger no matter how big the number field is.
How you should interpret a formal group law, even one so special as $\mathbf G_{\mathrm a}$, is up to you: different outlooks produce different theorems. Good luck.
