Let me answer the broad question first: depending on what you actually want to do, the barcode-type invariants extracted by topological data analysis could be quite useful in your work. And it doesn't take too much prior knowledge to use the TDA tools. For instance, if all you want to do is show that two datasets are qualitatively different, you can just compute their barcodes (I've written software to do this, as have others) and calculate the difference between them. It's easy, fast and free so why not try something orthogonal and complementary to your existing techniques?
The usual pipeline for TDA is as follows: starting with your data, you impose the structure of a filtered cell complex, compute the persistent homology, and output the barcode. The reason you may find it difficult to get precise answers to your questions is quite simple: everything depends on how the filtration is concocted! It is a bit of an art form to know exactly what to compute the persistent homology of, given the features that you actually care about.
Here's a typical TDA approach to your first question: let the dependent variable be $x$ and the independent variables $y_1,\ldots,y_n$.
Assume, by thresholding and binning if necessary, that each $y_j$ attains only finitely many states. Construct the flag complex on the $n$-partite graph whose vertex-bins correspond to values attained by the $y_j$s. Each simplex is weighted by the minimum $x$-value corresponding to the $y$-values fixed by its vertices. These weights give you a filtered simplicial complex, and generators of the $0$-dimensional persistence intervals of the super-levelset filtration tell you which configuration of $y$s correspond to which $x$ values.
Regarding your second question, a lot depends on what you mean by "predictions". For two examples of using persistent homology for predictions, consider Liz Munch's PhD thesis available here. It is possible to predict -- up to an extent -- coverage failure in sensor networks modeled as devices with a given failure probability. Perhaps a modification of this model could make predictions in situations that are of interest to you.