In some physics theories like string theory we have notion of spatially extended quantum objects - strings, membranes etc. Assuming that such objects exist, how would they appear in experiments ? More precisely are there any thought experiments for determining whether a quantum thing being observed has spatial extension or not, and if it has spatial extension then what is its dimension, topology etc.
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.
At the currently available energies, string will look like point particles. This is essentially by design: all the currently know elementary particles look like points so the next generation theory needs to predict that they will look like points.
Actually they needed to look like points at the energy scales of the 1970s and 80s, but they turned out to be so point-like on those scales that their extent is still unaccessible.