Roll coating is a vital industrial process used in printing, packaging, and polymer film production, where maintaining a uniform coating is critical for product quality and efficiency. This work models non-isothermal Carreau fluid flow between a rotating roll and a stationary wall under fixed boundary constraints to evaluate how non-Newtonian and thermal effects influence coating performance. The governing equations are transformed into non-dimensional form and simplified using lubrication approximation theory. Approximate analytical solutions are obtained via the perturbation technique, while numerical results are computed using both the finite difference method and the BVP-Midrich technique. Furthermore, Response surface methodology (RSM) is employed for optimization and sensitivity analysis. Analytical and numerical results show strong agreement (<1% deviation). The model predicts coating thickness $0.55\le \lambda \le 0.64$, power input $1.05\le P_w\le 1.99$, and separation force $0.91\le S_f\le 1.82$ for $0.1\le We\le 0.9$ and $0.01\le F\le 0.09$. Increasing $We$ enhances the coating thickness and power input but reduces velocity and separation force. The findings provide physical insight into elastic and viscous effects in roll coating, providing insight for optimizing coating uniformity, minimizing wear, improving industrial coating processes, and extending substrate lifespan.