general relativity - How does the string worldsheet affect the space-time in which they live?

I don't understand much about string theory and never really got much further past the Nambu-Goto action and very basic supersymmetry (SUSY) lectures in my undergraduate courses, but the only thing that stuck is that the string worldsheet lives in space-time.

So, if strings are allowed to interact only by string exchange (which amounts just to changing the genus of the worldsheet, that still keeps living in space-time), how do we go from that state of things where we have actions (that is, Polyakov) that depend on the metric properties of the worldsheet, and describe how the worldsheet evolve inside space-time as a fixed background, to strings-exchange-things-that-curve-spacetime-itself, understanding spacetime the very space those strings live within.

If strings are allowed to interact only via string exchange, does it mean that...

1. 
   
   

   strings are connected topologically to the space-time via strings? That would be weird, a two-dimensional surface connected to a four-dimensional one, but I guess it would be logically possible. But that suggests that strings basically live outside space-time and only touch it through its boundaries, so closed strings are out of this picture

   
   


2. 
   

   space-time is itself made of interacting strings? that would not make much sense to me. How would that preserve the traditional Lorentz invariance? Besides, string theory assumes the Lorentz invariance is exact

   
   


3. 
   

   strings have a mass density and just behave as classical general-relativity energy-momentum stress tensor densities? That would make sense, but then that would be openly cheating, since that would not explain how strings create gravity, since gravity would be added ad-hoc

   
   


4. 
   

   I'm out of options. How can strings interact with the space-time manifold?

   
   
   

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Edit

Thanks for the existing answers, but what I want, or what I was hoping is, if there is some diagrammatic/visual insight into the connection mentioned in the answers, say, the one from flat spacetime with graviton strings propagating that turns into an equivalent curved spacetime.