We investigate a discrete-time two-strain symbiotic epidemic model on complex networks with both random and long-range interactions. Our analysis examines how the co-infection recovery rate ($μ$), the long-range decay exponent ($α$), and the scale-free connectivity exponent ($γ$) shape epidemic persistence under cooperative dynamics. Comparison with a two-strain competition model shows how these parameters control strain dominance, coexistence, or extinction. The results demonstrate that contagion dynamics are strongly affected by environmental randomness and long-range couplings. In facultative symbiosis, the co-infection recovery rate undergoes a clear phase transition, separating persistence from extinction. In the competitive setting, regimes with $α< 2$ and $γ< 3$ markedly lower the epidemic threshold, allowing persistence even at small contagion rates ($σ$). Statistical analysis further reveals that $γ$ and $α$ exert pronounced, nonlinear, and time-dependent effects on strain survival.