In the frame of nonlinear Laplace- and Lame-type transmission problems, we consider the symmetric coupling as well as certain variants of the non-symmetric coupling of FEM and BEM. First, we introduce a unified framework, called implicit stabilization, that allows us to prove solvability and stability of the mentioned coupling methods, even in the presence of certain monotone non-linearities in the interior domain. This theory includes and even extends the well-known stability results for the symmetric coupling and extends the existing theory for non-symmetric couplings. Unlike prior work, we remove any assumption on the given mesh like, e.g., being sufficiently fine along the boundary. Moreover, our work provides the first mathematical stability results for non-symmetric FEM-BEM couplings and nonlinear problems. Second, we consider adaptive mesh-refining methods. Mainly, we are interested in the convergence of the corresponding adaptive algorithm steered with the weighted-residual error estimator, where we give the first mathematically reliable convergence results. Additionally, we analyze a gradient recovery estimator (ZZ-estimator) in terms of reliability and efficiency. Third, we consider preconditioning of Galerkin matrices arising from BEM operators on locally refined triangulations. It is well-known that the condition number of the unpreconditioned Galerkin matrices grows if the mesh is refined, and additionally hinges on the ratio between the maximal and minimal mesh-size. Within the framework of additive Schwarz methods, we introduce a local multilevel diagonal preconditioner for the hypersingular operator in 2D and 3D, and for the weakly-singular integral operator in 2D and prove that the condition numbers of the preconditioned systems are optimal, i.e., independent of the mesh-size. Fourth, we analyze a (2x2)-block-diagonal preconditioner for the Galerkin matrix of the coupling systems. The two blocks correspond to preconditioners for the FEM part and the BEM part, respectively. Provided that we have optimal preconditioners for the FEM part and the BEM part, we prove optimality of the block-diagonal preconditioner for symmetric as well as non-symmetric coupling methods. In particular, such a result applies for the local multilevel diagonal preconditioners and provides, at least for 2D problems, optimal preconditioners for the FEM-BEM couplings under consideration. Throughout, we give various numerical examples that underline our theoretical findings.