About showing algorithmic gap instance for the Goemans-Williamson SDP Using usual notation we have,

$SDP(G) \geq OPT(G) \geq Alg_{GW}(G) \geq \alpha_{GW} SDP(G) \geq \alpha_{GW} OPT(G)$

where we mean, 

$SDP(G)$ = The maximum value that the SDP finds of the objective function $\sum_{(u,v) \in E} w_{uv}( 1-\vec{y_u}.\vec{y_v})/2$  (one unit vector $\vec{y_v}$ in $\mathbb{R}^d$ for each vertex $v$ for some $d$) And $w_{uv}$ are the given edge weights such that $\sum_{(u,v) \in E} w_{uv} =1$ 

$OPT(G)$ = The actual size of the max-cut for the graph

$Alg_{GW}(G)$ = The value of the max-cut returned by the randomized rounding prescription of Goemans-Williamson

$\alpha_{GW}$ = the famous constant of $\sim 0.878$ 


If say someone shows that for every $\epsilon$ there is a graph $G(\epsilon)$ such that $(\alpha_{GW}+\epsilon)OPT(G(\epsilon)) \geq Alg_{GW}(G(\epsilon))$ then does this imply that this is a class of graphs on which the algorithm is performing as bad as it could? 
Why does showing the existence of such a family of graphs as above also  necessarily imply that on this family $SDP(G(\epsilon)) = OPT(G(\epsilon))$ ?
A graph for which $Alg(G) = \beta OPT(G)$ is, by definition, a graph on which the algorithm has approximation ratio $\beta$. You can use this insight to answer the first question.

As to your second question, suppose that $Alg(G) \leq (\alpha_{GW}+\epsilon) OPT(G)$. Since $Alg(G) \geq \alpha_{GW} SDP(G)$, we deduce that
$$
\alpha_{GW} SDP(G) \leq Alg(G) \leq (\alpha_{GW}+\epsilon) OPT(G).
$$
Therefore $SDP(G) \leq (1+O(\epsilon)) OPT(G)$.