Why render the kinetic part of a vector field (other than the photon) $U(1)$ symmetry? I have seen a number of papers, for example the ones considering dark vector field, in which the kinetic part of the vector field is rendered $U(1)$ symmetry, i.e. they have the form known from Q.E.D. I wonder why one can't write something else, for example $$0.5 * \partial_\alpha A_\mu \partial^\alpha A^\mu~?$$
Why consider $F_{\mu\nu} F^{\mu\nu}$ form so abundantly?
 A: The most general first-order kinetic term you could write down for a vector field $A_\mu$ is
$$ S_\text{kin}[A] = \int \left( c_1 \partial_\mu A_\nu \partial^\mu A^\nu + c_2 \partial_\mu A_\nu \partial^\nu A^\mu\right) \mathrm{d}x.$$
but this is in general a bad action because the corresponding energy is not bounded from below: All terms that involve a $\partial_0 A_\mu$ must have the same sign so that the resulting Hamiltonian can be bounded from below, but the $(\partial_0 A_0)^2$ term here has a different sign than the $(\partial_0 A_i)^2$ terms. So we choose $c_1 = -c_2$ to eliminate the $(\partial_0 A_0)^2$ and obtain a Hamiltonian bounded from below, but then up to a rescaling we can choose $c_1 = -c_2 = \frac{1}{2}$ and you can check that this then is just the standard Maxwell kinetic term $\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$.
Also, note that this argument works equally well in the presence of a mass term $\frac{m^2}{2}A_\mu A^\mu$, so it has nothing to do with $\mathrm{U}(1)$ symmetry - the $F_{\mu\nu}F^{\mu\nu}$ is simply the only consistent choice you can make in order to have bounded kinetic energy for both massive and massless vector fields.
