MADMAX (the MAgnetized Disk And Mirror Axion eXperiment) is a novel approach to search for Axion Dark Matter in the mass range 40-400 $\mu$eV/c$^2$. It utilizes the principle of the dielectric haloscope, a system composed of several parallel dielectric disks, a mirror, and a receiver. This so-called booster system is placed within a magnetic field and a cryogenic environment. We investigate the electromagnetic properties, systematic effects, and the thermal noise of several small prototype versions of the MADMAX experiment, including a proof of principle booster setup, and small (~10 cm) diameter tube designs. Measurement of electromagnetic signal reflection, dielectric disk placement precision, antenna designs, and thermal background signals are studied and the results are presented.