In the conditions of an energy crisis, an important issue is the increase in energy efficiency and productivity of welding and hardfacing processes. The article substantiates the perspective of using exothermic additives introduced into core filler for flux-cored wire arc welding processes as a relatively cheap additional heat source, reducing energy consumption when melting filler materials, and increasing the deposition rate. The mixture design (MD) was selected as the design method to optimize the average values of current and voltage, as well as arc stability parameters depending on core filler composition. This article studies the influence of the introduction of exothermic addition (EA), as well as the ratios CuO/C and CuO/Al on arc stability for the FCAW S process. Parameters characterizing arc stability were determined using an oscillograph, and from the obtained oscillograms, an analysis was conducted on arc voltage and welding current signals during flux-cored arc welding. It was determined that various methods can be used to evaluate arc stability, which can be divided into two groups: graphical (current and voltage cyclograms, box plots with frequency histograms, ellipse parameters plotted on current, and voltage cyclograms) and statistical (standard variation and coefficients of variation for welding current and arc voltage). In this paper, a comprehensive evaluation of arc stability depending on the composition of the cored wire filler was carried out. It was determined that the most stable current parameters were observed for the flux-cored wire electrode with an average exothermic addition content at the level of EA = 26.5–28.58 wt.% and a high carbon content (low values of CuO/C = 3.75). Conversely, the lowest values of arc stability (CV(U) and Std(U)) were observed during hardfacing with a flux-cored wire electrode with a high CuO/Al ratio ≥ 4.5 and a content of exothermic addition in the core filler below the average EA < 29 wt.%. Mathematical models of mean values, standard deviation, coefficient of variation for welding current, and arc voltage were developed. The results indicated that the response surface prediction models had good accuracy and prediction ability. The developed mathematical models showed that the ratio of oxidizing agent to reducing agent in the composition of exothermic addition (CuO/Al) had the greatest influence on the welding current and arc voltage characteristics under investigation. The percentage of exothermic mixture in the core filler (EA) only affected the average welding current (Iaw) and the average arc voltage (Uaw). The graphite content expressed through the CuO/C ratio had a significant impact on welding current parameters as well as the coefficient of variation of arc voltage (CV(U)). Two welding parameters were selected for optimization: the mean welding current (Iaw) and the standard deviation of arc voltage (Std(U)). The best arc stability when using exothermic addition CuO-Al in the core filler was observed at CuO/Al = 3.6–3.9, CuO/C = 3.5–4.26, and at an average EA content of 29–38 wt.%. The significant influence of the CuO/Al and CuO/C ratios on arc voltage parameters can also be explained by their impact on the elemental composition of the welding arc (copper, cupric oxide (CuO), and Al2O3). The more complete this reaction, the higher the amount of easily vaporized copper (Cu) in the arc plasma, enhancing arc stability. The influence of core filler composition on the microstructure of deposited metal of the Fe-Cr-Cu-Ti alloy system was investigated.