More precisely are there any thought experiments for determining whether a quantum thing being observed has spatial extension or not,
You are in effect asking if there exists a theoretical program to check whether the proposed strings have any experimental signatures that verify their existence unequivocally, which verification would also include the fact that the string is different than a point particle.
This is the whole project of string theory and string phenomenology as a "thought experiment" and most of the experimental program of the LHC tries to determine which phenomenological string model is physically realized, if at all.
Suppose that the LHC and its successor, possibly the ILC find a specific string phenomenological model is verified by the measurements, i.e. the new resonances found, the branching ratios and symmetries observed fit very well with one of the proposed models, then the model will be validated, and the premises of the model will be validated, i.e. the supposition of string dimensions in that model.
One has to realize that the grand majority of our knowledge of the behavior of the microcosm comes from measurements in the macrocosm that verify the hypothesis of the theoretical models. That we all accept that an electron is a point particle does not come because we have taken a ruler and measured its dimensions. It comes because the hypothesis that it is a point particle fits perfectly the accepted models for the microcosm at the moment, which is the so called standard model of particle physics. If a hypothesis with strings fits all new and old data better then the strings are as real as points, as far as the microcosm goes.
The resonance and scattering spectra at high energies are the tools to see the dimensions of objects in the microcosm through the use of strict mathematical theories.
and if it has spatial extension then what is its dimension, topology etc.
It will be the dimension, topology etc assumed in the model that will be validated, in the future, and of course if there exists such a validated outcome. Otherwise new theories will be pursued.
In other words our "rulers" have become extremely layered and complex.
This post imported from StackExchange Physics at 2014-03-24 04:33 (UCT), posted by SE-user anna v