We discuss a fictitious domain approach with distributed Lagrange multiplier (FDDLM) for interface problems with discontinuous coefficients, developed within the framework of unfitted immersed methods. The proposed formulation leverages a discontinuous Lagrange multiplier to achieve enhanced local enforcement of distributed constraints. We present a new family of finite element methods that ensures the well-posedness of the discrete problem. Additionally, we introduce a novel a posteriori error estimator for FDDLM, rigorously proven to be both reliable and efficient. This estimator guarantees optimal convergence, even in the presence of singular solutions or complex geometries. While this study focuses on stationary cases to develop and refine the method, it establishes a milestone for tackling dynamic FSI problems. The method's versatility and robustness are showcased through its application to elliptic interface problems, Stokes interface problems, as well as coupled Stokes-elliptic interface problems.