Why isn't the contrapositive of the E proposition always valid? If I have, for example, a statement: "No dog is a whale" I can take the obverse and say: "All dogs are not whales" And turn this into an implication: "If it is a dog, then it is not a whale" And because the contrapositive is logically equivalent, I can say: "If it is a whale, then it is not a dog" So then does this mean the final statement is not always true? Or am I misunderstanding the fact that an E proposition doesn't necessarily have a contrapositive?

In your example, you do not use the procedure described in the article you mentioned). After obversion, you obtain "All dogs are not whales". If you want to write the converse of this statement, you should switch "dogs" and "not whales". You would obtain "All animals which are not whales are dogs", which is not true.

I now write your example with mathematical notations. Let $D$ be the set of dogs. Let $W$ be the set of whales. We assume that "No dog is a whale": $$\
eg (\exists d \in D,\ d\in W).$$ We then have "All dogs are not whales": $$\forall d \in D,\ d \
otin W.$$ This implies that "If it is a dog, then it is not a whale": $$\forall d,\ ( (d \in D) \Rightarrow (d \
otin W) ),$$ and then "If it is a whale, then it is not a dog": $$\forall d,\ ( (d \in W) \Rightarrow (d \
otin D) ).$$ Even if $D$ and $W$ are other sets, the final statement will still be true.