Why in some cases $0\alpha$ component of stress-energy tensor don't form 4-vector? In electrodynamics there is Poynting vector and energy density, which refer to $0\alpha $ components of stress-energy tensor, don't create 4-vector. Analogous situation with mass density and mass current density in general relativity. So why don't they form 4-vector (with what property is it connected)?
 A: Maybe I did not understand your question. However, referring to the stress-energy tensor, the four-vector you want is:
$$S_\nu = \int_{\Sigma_t} T_{0\nu} d^3x$$
where $\Sigma$ is the rest 3-space $x^0=t$ of Minkowski reference frame with coordinates $x^0,x^1,x^2,x^3$. In view of the relation $\partial_\mu T^{\mu \nu}=0$ the definition of $S$ does not depend (1) on the used Minkonskian reference frame,(2) on  the value of $t$, (3) it defines a four-vector, and (4) it is conserved: $\partial_\mu S^\mu=0$.
All that is true if the region in spacetime  where $T \neq 0$ has compact intersection with every such $\Sigma_t$ (it is sufficient that it happens for one of them to be valid for all and for all reference frames). This condition can however be weakened.
The proofs of all I wrote easily arise from the (four-dimensional) divergence theorem.
ADDED NOTE. All the above discussion holds in Minkowski spacetime, in curved spacetime the picture becomes more complicated.  In a globally hyperbolic curved spacetime, a conserved current is obtained from a symmetric conserved stress energy tensor in the presence of a Killing vector $\xi$. Then $J^\mu:=\xi_\nu T^{\nu\mu}$ verifies $\nabla_\nu J^\nu=0$. In this case $$Q:= \int_\Sigma J^\mu n_\mu d\mu$$ is a conserved "charge" independent from the spacelike Cauchy surface $\Sigma$. Above $\mu$ is the natural measure induced on $\Sigma$ by the metric of the spacetime and $n$ the unit normal co-vector. The components of $S^\mu$ above have this structure, $\xi^{(\mu)}_\nu$ being the constant vector fields in Minkowski spacetime associated with a Minkowskian reference frame.
