In the beginning of string theory textbooks, strings are said to live in a background "target" space time. They then propagate through this space time. Strings also have a spin 2 ("graviton") mode, and can scatter off of each other. So in one sense, it seems like strings live "in" space time. Gravitational waves can then be thought of as coherent states of "string plane waves," constructed from asymptotic states.
But somehow, don't these strings also affect the space time they are in? It is often said that string theory has "no free parameters," but that different laws of physics (low energy effect QFTs) are given by different compactifications of extra dimensions. In some fuzzy sense, somehow these different compacifictions are minima of some potential energy (although I don't see how). What do these compactifications have to do the strings?
In other words, it seems as though a space time manifold is an input to your theory, and strings propagate in it. However it also seems like somehow the strings can select which space time they are in, because there are different possible compacifications of extra dimensions. So do strings live in space time, or do they "make" space time, somehow?
(I am a person who knows QFT but not much about string theory. I welcome any math used in an answer.)