The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the Ψ-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the Ψ-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann-Liouville-type and Caputo-type cases of these operators. Furthermore, we study some non-linear fractional differential equations involving Ψ-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle and proving stability results by using Grönwall type inequalities.