1. → I modeled a plate supported on its boundaries, and the deflections do not match the ones I calculated based on analytical formulas by about 3%. Why?
The finite element method is an approximate method. Its accuracy depends on the mesh refinement and type and geometry of finite elements used. In some cases for the regular beam or truss element the results may be the exact ones due to the degree of the displacement shape function adopted in the formulation of the element. For surface elements, the results are always approximate, and you can evaluate the precision of the results only by comparing them to ones obtained on a model with a more refined mesh, or based on modeling experience.
2. → What type of finite element to use to model a reinforced concrete water tank embedded into the ground?
Usually the use of shell elements is required. The embedment can be modeled using surface supports with appropriate stiffness (determined based on soil tests).
3. → How can I model semi-rigid connections within a steel structure?
You can use spring elements. The stiffness characteristics should be assigned according to the connection models available in the design code or determined experimentally.