While measuring the ionic conductivity of Solid electrolyte Zimg and Zreal are way off. Can anyone explain the reason behind this? and also the low frequency tail is never crossing the X axis(Zreal).
Zreal and Zimag are almost never the same, and if they are it is coincidence as they are representative of different phenomena. Zreal is "resistive" and has to do with, in your case, ionic conductivity (up until depletion effects show up) and in most EIS testing, charge transfer resistances. Zimag is out of phase, and most often relates to capacitive effects. It can also come up with diffusion effects, which I'm guessing is your thought, but that diffusion effect is actually from depletion of [electroactive species] at the electrode surface and not ions moving through solution. And it can be related to inductive effects--mostly at high frequencies in low Z systems where it is mostly from cable inductance, but also at low frequencies with adsorbing species.
Your other question is harder to answer but a couple possibilities are: you are not going to a low enough frequency, or the impedance is such that it will never cross (depletion/diffusion and capacitive effects).