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The answers are usually always valid. What may or may not be valid are your assumptions about the underlying model. Also, the number of times the results should be similar depends on the number of possible outcomes and the variability in the outcomes. For example, if you spin a fair spinner with 12 equal segments, then the probability of similar results is less than likely.
What do you mean by this? 36 inches? Feet? Centimeters? You need to be a bit more specific to receive a valid answer.
The main difference between the Rayleigh-Ritz method (RRM) and the finite element method lies in the definition of the basis functions. For FEM, these are element-related functions, whereas for RRM these are valid for the whole domain and have to fit the boundary conditions. The Rayleigh-Ritz method for homogeneous boundary conditions leads to the same discretized equations as the Galerkin method of weighted residuals.
A scientific theory is a hypothesis that has been repeatedly tested and still seems to be valid. These theories are tested by using the scientific method.