Terpolymerizations of newly synthesized ethylene (E), vinylcyclohexene (VCH), and 1-hexene were carried out with symmetrical metallocene catalysts rac-Me2Si(2-Me-4-Ph-Ind)2ZrCl2 (catalyst A) and rac-Et(Ind)2ZrCl2 (catalyst B). X-ray diffractometry (XRD), scanning electron microscopy (SEM), differential scanning calorimetry (DSC), high-temperature gel permeation chromatography (GPC), and nuclear magnetic resonance (NMR) spectroscopy were used to evaluate the behavior and microstructure of the polymers. The activity of catalyst B was 1.49 × 106 gm/mmolMt·h), with a Tm of 73.45 (°C) and ΔHm of 43.19 (J/g), while catalyst A produced first higher 1-hexene, 19.6 mol %, and VCH contents with a narrow molecular weight distribution (MWD). In previous reports, ethylene propylene monomer dienes (EPDM) had a low content and were used for dielectric and insulating properties with nanomaterials. Second, this paper presents a kind of elastomeric polymers based on E/1-hexene and VCH with a high dielectric constant (k = 6–4) and mechanical properties. In addition, low dielectric loss suggests the suitable application potential of these polymeric materials for the fabrications of capacitors. Also, this work reveals that these polymers can be a better candidate for high-voltage electrical insulation due to their enhanced dielectric, mechanical, and thermal characteristics. To examine the insulating property, the interface characteristics of the polymer were evaluated using electrochemical impedance spectroscopy (EIS) with a frequency range of 1 × 105–0.01 Hz and an amplitude of 5.0 mV. EIS is an effective method to investigate the polymers’ interfacial electron transfer characteristics. The EIS Nyquist plot showed high Warburg impedance features in the low-frequency domain with straight lines without a semicircle, suggesting that the property of the polymer owing to the high electrical resistance and poor conductivity for ionic kinetics in the electrolyte may have surpassed that of the semicircle. Although the slope of low frequencies in polymers holding potent exoelectrogenic bacteria (Shewanella oneidensis MR-1) as a charge carrier in the electrolyte could significantly reduce the Warburg resistance, it still could not improve the conductivity, which demonstrated that the external charge supply could not alter the insulating property in the used polymers.