I can’t teach a whole mechanics course in the comments of one post on a finite element software forum posts, but it might be worth clarifying one or two of your statements above to add a bit of “application specific” expertise here. Hopefully this is helpful!:
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Moments in a beam model actually only arise from accounting for the various stresses through a beam’s cross-section. In a 3D solid mechanics formulation (such as the one you linked to in Jorgen’s excellent dolfinx tutorial), we don’t need to account for moments. Their effect is “built into” the 3D solid mechanics formulation.
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The u and v you mentioned are the displacement trial and test functions, used to solve for what the displacement field looks in a solid model given specific loads and boundary conditions. \nu is a material parameter commonly used in mechanics for relating stress to strain. There are many “constitutive models”, but a common one is linear elasticity, which so happens to require only two material parameters to relate stresses to strains. One instance is the pair of E and \nu, the Elastic modulus and poisson ratio. There are other forms such as Lamé parameters, \lambda and \mu, which are used in the FEniCSx tutorial. You can see in this wikipedia page how these parameters relate to each other.
I would suggest having a look at https://www.continuummechanics.org/ and http://solidmechanics.org/ as they are both free and high quality resources to learn a bit more about solid mechanics.
P.S. if you are interesting in only transverse deflection calculations and extracting moments/forces, there are a few implementations of beam models on the forums, such as here.