Your answer to this question is the science of Di-electric analysis (DEA), so a short answer can't really do it justice here, but you will be able to find many answers with a search for DEA theory. Maybe helpful though to say that, DC conductivity shows that ions or electrons may freely pass through the entire structure of the sample, whereas poor DC conductivity but good AC conductivity would indicate only local ionic/electron mobility.
Apologies that I'm not an expert on this to answer the other questions.
In order to determine the AC conductivity, you typically require to measure the response of your specimen over an applied sinusoidal current wave of differential frequency. This could be accomplished by complex impedance (EIS) measurements. With an appropriate equivalent circuit model (ECM) fitting, you should be able to determine the values of individual circuit components and resistivity depending upon the shape factors. The reciprocal of determined resistivity will be DC conductivity.
Have a look at the following relevant literature(s)-
Article AC impedance spectroscopy and conductivity studies of Dy dop...
Article Fast ionic conduction in tetravalent metal pyrophosphate-alk...