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Many sources state that one of the primary rules for molecular structures is that the least electronegative element is in the center. However, none of them explain why this is the case.
Is this because it is most conducive to sharing electrons and forming bonds, or is it the most stable, or is there a better reason?
Also, why are many molecules symmetric. Is there a reason why symmetry in nature and in chemistry is preferred over non-symmetry?
Why are least electronegative elements usually in center of lewis structure?
If there is a "central atom", then there are multiple (more than 1) bonds to the central atom. If the central atom is relatively electropositive, then it will be better able to share its electrons and form bonds with other atoms, at least more so than an electronegative central atom would. Also, electronegative atoms tend to carry multiple lone pairs of electrons. If this electronegative atom and all of its lone pairs were at the center of the molecular structure, then we would have many more destabilizing (lone-pair - bonding pair) electron-electron repulsions, then if all of these lone pairs were on the periphery of the molecule.
why are many molecules symmetric. Is there a reason why symmetry in nature and in chemistry is preferred over non-symmetry?
The symmetric arrangements (linear, tetrahedral, octahedral, etc.) minimize destabilizing bonding electron - bonding electron repulsions. Remember methane is a perfect tetrahedron - that is the lowest energy arrangement because all of the bonding electrons are as far apart from one another as possible in this geometry.
Is this because it is most conducive to sharing electrons and forming bonds, or is it the most stable, or is there a better reason?
Your first rationalization is a good one. That is the likely reason the least electronegative element generally becomes the central element. That being said, there are exceptions. One being water. Oxygen is much more electronegative than hydrogen, which suggests that hydrogen should be the central atom, but the hydrogens are clearly peripheral atoms. This stems from the limitation that hydrogen can only form one sigma bond. That being said, hydrogen has also been found in unconventional bonding arrangements (see diborane for an example).
Also, why are many molecules symmetric. Is there a reason why symmetry in nature and in chemistry is preferred over non-symmetry?
Symmetrical conformations of molecules may or may not have lower energies than their asymmetrical counterparts. Consider the ternary interhalogen, $\ce{IBrF2}$. The symmetrical conformation is likely lower in energy than the asymmetrical conformation (draw out the Lewis structures to see).
As we can see, in the symmetrical conformation (left), there are only 90 degree fluorine/lone pair interactions, as opposed to the conformation on the right, in which there is a fluorine/lone pair interaction in addition to a bromine/lone pair interaction. The equatorially placed atoms have been left out of this consideration as are ~120 degrees apart from the lone pairs and thus their interactions are not as significant as the 90 degree axial/lone pair interactions.
That being said, there are myriad exceptions. Considered the eclipsed (left) and non-eclipsed (right) conformations of 2,3-dibromobutane. The eclipsed conformation is symmetric (it is also meso; all meso compounds are symmetrical but not all symmetrical compounds are meso). The non-eclipsed conformation is asymmetric. However, the non-eclipsed conformation is energetically preferred since eclipsing the bromines places them closer together, which creates more van der Waals repulsions.
Additionally, symmetry is a complicated topic in chemistry, especially with respect to more complex molecules and entities.
