This article presents a mathematical and computational analysis of the adjoint problem approach for parabolic inverse coefficient (or inverse heat conduction) problems based on boundary measured data. In Part I the mathematical analysis is given for three classes of typical inverse coefficient problems with various Neumann or/and Dirichlet types of measured output data. Although all three types of considered inverse coefficient problems are severely ill-posed, comparative numerical analysis show that the ill-posedness depends also on where the Neumann and Dirichlet conditions are given: in the direct problem or as an output data. For all these types of inverse problems the integral identities relating solutions of direct problems and appropriate adjoint problems solutions are derived. These integral identities permit proof of monotonicity, Lipschitz continuity, and hence invertibility of the corresponding input-output mappings. Based on these results solvability of all three types of inverse coefficient problems are proved. The degree of ill-posedness of inverse problems is demonstrated on numerical test examples.