Molecular Orbital Mixing to get the Molecular Orbitals of a large molecule
Running the calculation without imposing symmetry constraints should give the same ground energy as one with them because the true electronic ground state of the system (within the Born-Oppenheimer approximation) always has the same symmetry as the nuclei. The variational principle always puts an upper bound on your ground state energy, so in principle it will work just as well. Using symmetry, however, could speed up your calculation significantly because the symmetry operators commute with you Hamiltonian. This gives you an additional set of constraints on the coefficients. It is worth mentioning that single reference methods like Hartree-Fock and DFT will sometimes give lower ground state energies without symmetry constraints because the true ground state is a symmetric superposition of several configurations, but the single reference methods can only examine a single one.

For polymer molecules and other 1D infinite systems you would use periodic boundary conditions. Just make the box really large in the two dimensions you don't want to be periodic (so the system doesn't 'see' itself) and only evaluate non-gamma k-points along the periodic vector.