The least-squares finite element method (LSFEM) is attracting interest in recent years as an alternative method for solving partial differential equations (PDEs) related to mechanical problems. The method offers several advantages over the well-known principles of Galerkin variational methods, such as a unified mathematical procedure in constructing first-order systems for all types of PDEs. Moreover, it leads to positive definite and symmetric system matrices, and it provides an a posteriori error estimator without additional cost. These advantages also apply to differential equations with non-self-adjoint operators such as the incompressible Navier-Stokes equations. In this work, LSFEM is applied to the continuous incompressible Stokes and Navier-Stokes equations, and the small strain elastodynamics equations. The objective is to investigate the accuracy and performance of various formulations proposed for fluid and solid problems. In addition to discussing implementation aspects, various numerical tests are investigated. Simulation of fluid flow through 3D porous structures is one application area considered in this work. Furthermore, adaptive meshing strategies are investigated. A particular focus is on formulations in terms of stresses and velocities as primary fields that can be used in the least-squares FEM-based fluid-structure interaction (FSI) approach. The proposed LSFEM-FSI approach is based on the idea of solving the interaction monolithically, where the interaction conditions are inherently satisfied by the conformal discretization of the stresses and velocities in suitable Sobolev spaces. Numerical tests of the LSFEM-FSI in the regime of small strains are provided.