Are there any models in string theory which are background independent?
Non-perturbatively, it is conjectured that there is a unique, underlying framework (M-theory), and all existing calculations in string theory are approximations that in principle should be derivable from this larger framework. So there are not different string theory models, but rather one unified non-perturbative framework. (However, no one knows precisely what this framework is, in the sense that no one can write down a fully non-perturbative definition of M-theory and derive all known string theory results from it).
This non-perturbative framework should be "background independent," in the sense that one should be able to solve its equations of motion to find classical backgrounds, and perturb around these backgrounds.
In practice, since we don't know how to formulate M-theory, many calculations either assume a particular background and work perturbatively, or solve for a background in some limit where the theory is well understood (eg supergravity). This practical limitation is not an "in principle" problem though -- it is not as if M-theory has some assumed, fixed background built into it.
If there are, would this mean that these models could be built in any number of dimensions? (Instead of assuming a fixed number of dimensions as it is usually done in string theory)?
No. The dimension of spacetime is fixed in string theory, in order to have a consistent Lorentz-invariant, unitary quantization of the theory. (although, weirdly, in different limits the apparent number of spacetime dimensions is different)
However the extra dimensions can be compactified, so in practice the low energy physics that we observe is 4-dimensional.
And finally, if there could be string theories which are background independent, would this mean that these models would assume no symmetries, as physicist Lee Smolin says 1:
We have posited that the fundamental theory is background independent, which means there are no symmetries
Would this mean that background independent string theories would not assume any symmetry as fundamental (implying that global, local, gauge or even Lorentz and CPT symmetries are not fundamental)?
To be honest I don't really understand what this quote means. However it is a generic property of quantum gravity that there are no global symmetries. Gauge symmetries are allowed (and indeed necessary if one is going to have a hope of embedding the Standard Model into string theory). They can be implemented in string theory by, for example, having multiple D-branes stacked at the same position. Having said that, gauge symmetries are really redundancies of description, not fundamental symmetries, and it should be possible (albeit probably much more complicated) to formulate the theory without gauge symmetries.