In physics you frequently find that a simple theory is part of a bigger, more complex theory, and that in turn is part of an even bigger, even more complex theory. For example non relativistic quantum mechanics (e.g. Schrodinger's equation) is a low speed limit of quantum field theory, and quantum field theory is a low energy limit of string theory.
When solving a problem you always choose the simplest theory you can get away with. You wouldn't attempt to use string theory to calculate the structure of a hydrogen atom because the calculation would be hopelessly complex. However we know QFT is a low energy limit of string theory and we can use QFT to calculate the structure of a hydrogen atom. Actually even non-relativistic QM is adequate for most purposes.
Given the above, the key bit of your question is:
has it been shown yet that string theory becomes quantum mechanics at certain dimensions?
and the answer is that yes, string theory does reproduce QFT at low energies. I don't think a model has yet been constructed that fully reproduces the Standard Model, but it's widely believed that it's possible and that we simply haven't found it yet.