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We report four new detections of 21-cm absorption from a systematic search of 21-cm absorption in a sample of 17 strong (rest equivalent width, W_r(MgIIλ2796)≥1Å) intervening MgII absorbers at 0.5<z_abs<1.5. We also present 20-cm milliarcsecond scale maps of 40 quasars having 42 intervening strong MgII absorbers for which we have searched for 21-cm absorption. These maps are used to understand the dependence of 21-cm detection rate on the radio morphology of the background quasar and address the issues related to the covering factor of absorbing gas. Combining 21-cm absorption measurements for 50 strong MgII systems from our surveys with the measurements from literature, we obtain a sample of 85 strong MgII absorbers at 0.5<z_abs<1 and 1.1<z_abs<1.5. We present detailed analysis of this 21-cm absorption sample, taking into account the effect of the varying 21-cm optical depth sensitivity and covering factor associated with the different quasar sight lines. We find that the 21-cm detection rate is higher towards the quasars with flat or inverted spectral index at cm wavelengths. About 70% of 21-cm detections are towards the quasars with linear size, LS<100pc. The 21-cm absorption lines having velocity widths, ΔV>100km/s are mainly seen towards the quasars with extended radio morphology at arcsecond scales. However, we do not find any correlation between the integrated 21-cm optical depth, ∫τdv, or the width of 21-cm absorption line, ΔV, with the LS measured from the milliarcsecond scale images. All this can be understood if the absorbing gas is patchy with a typical correlation length of ∼30-100pc. We confirm our previous finding that the 21-cm detection rate for a given optical depth threshold can be increased by up to a factor 2 by imposing the following additional constraints: MgII doublet ratio <1.1, W(MgII)/W(FeII)<1.47 and W(MgI)/W(MgII)>0.27. This suggests that the probability of detecting 21-cm absorption is higher in the systems with high N(HI). We show that within the measurement uncertainty, the 21-cm detection rate in strong MgII systems is constant over 0.5<z_abs<1.5, i.e., over ∼30% of the total age of universe. We show that the detection rate can be underestimated by up to a factor 2 if 21-cm optical depths are not corrected for the partial coverage estimated using milliarcsecond scale maps. Since stellar feedback processes are expected to diminish the filling factor of cold neutral medium over 0.5<z<1, this lack of evolution in the 21-cm detection rate in strong MgII absorbers is intriguing. Large blind surveys of 21-cm absorption lines with the upcoming Square Kilometre Array pathfinders will provide a complete view of the evolution of cold gas in galaxies and shed light on the nature of MgII systems and DLAs, and their relationship with stellar feedback processes.