When can partial spectral sparsifiers be combined?
Regarding your 1st question: no, this would not hold in general.

Rather (answering your 2nd question), the right way to interpret the approach by Spielman and Teng is as follows: let $G'$ be a vertex-induced subgraph of $G$, and $H'$ a spectral sparsifier of $G'$. Then the graph $G-G'+H'$ (replace all edges in $G'$ by those in $H'$) will be a spectral sparsifier of $G$. This easily follows by combining the Laplacians of $G$ and $H'$. As a result, we can sparsify $G$ by subsequently sparsifying subgraphs.

And finally, you are asking about ways to amplify the success probability of a sparsification algorithm. For spectral sparsification, there is a rather easy way: construct one with constant success probability, check whether it is indeed a spectral sparsifier (this amounts to checking whether a certain matrix is PSD, which can be done efficiently), if not, reiterate. After $O(\log(1/\epsilon))$ iteration this outputs a spectral sparsifier with probability at least $1-\epsilon$. Note that this cannot be done for say cut sparsification, as there is no efficient "verification procedure" to check whether a graph is a cut sparsifier of another graph.