Why would the normal force change between the different iterations?
Because a +Y lifted off, or perhaps sat down. Each iteration is independent.
How to use the information obtained from the status of the non-converging nodes in solving convergence problems?
1) The system parameters are used to define the global stiffness matrix [K] and load vector {f}. This defines the system of equations [K]{x} = {f}.
2) In a system with non-linear boundary conditions, each load case must undergo an iteration process. In this process, the above system of equations is solved for {x}. Then, at each non-linear boundary condition, the status is checked. If the boundary condition changed (a +Y lifted off, a gap opened or closed, etc), then that DOF in the stiffness matrix [K] is altered. When all changes to [K] have been made, the next iteration for that load case begins. This process is repeated until all boundary conditions are within the convergence tolerances.
3) The only convergence tolerances users can control are for friction and large rotation rods. Items like +Y supports, gaps, and soil restraints are either on-off, or yielded-nonyielded.
4) CAESAR II doesn't have an iteration limit. If the job doesn't converge, we don't give an answer. We feel no answer is better than a wrong answer.
If during the solution, you click the [F2] key, CAESAR II will give you a list of the restraints that are not converged at that time. Changing characteristics of any of these restraints may allow the job to converge. Details of what this report contains are discussed in my post above.
For jobs with friction, please read the section on friction in the Technical Reference Manual. Additionally, the magnitude of the "coefficient of friction" can be changed globally for all restraints using the "load case options" tab of the static load case editor.
I can not say how any one specific boundary condition change will affect the system of equations in [K], and the subsequent matrix decomposition. When you encounter this behavior, all you can do is play with the system, using the list of non-converged restraints as a guide.