How does treewidth behave under graph minor operations?
The decrease in tree width of any of the three operations will be 0 or 1.  The proof is simple (I will take the case of the vertex deletion).

Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.

Let $T'$ be an optimal tree decomposition of $G'$, with $\text{tw}(G') = w'$.

Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$.  Clearly, $T$ is a tree decomposition for $G$ with width $\text{tw}(G) \leq \text{tw}(T) = 1 + \text{tw}(G') = 1 + w'$.