  We present the updated census and statistics of Lyman-$\alpha$ emitting long
gamma-ray bursts host galaxies (LAE-LGRBs). We investigate the properties of a
sub-sample of LAE-LGRBs and test the shell model commonly used to fit
Lyman-$\alpha$ (Ly$\alpha$) emission line spectra. Among the LAE-LGRBs detected
to date, we select a golden sample of four LAE-LGRBs allowing us to retrieve
information on the host galaxy properties and of its interstellar medium gas.
We fit their Ly$\alpha$ spectra using the shell model, and constrain its
parameters with the observed values. From the comparison of the statistics and
properties of LAE-LGRBs to those of LAE samples in the literature, we find
evidences of Ly$\alpha$ suppression in dusty systems, and a fraction of
LAE-LGRBs among the overall LGRB hosts lower than that found for Lyman-break
galaxy (LBG) samples at similar redshift range. However, we find that LAE-LGRBs
are representative of Ly$\alpha$ emission from the bulk of UV-selected galaxies
at z~2. We find that the golden sample of LAE-LGRBs are complex systems
characterized by multiple emission blobs and by signs of possible galaxy
interactions. The fitting procedure fails in recovering the HI column densities
(NHI) measured from the afterglow spectra, and the other properties described
by the shell-model parameters in the cases with very high NHI. The afterglows
of most LGRBs and LAE-LGRBs show high NHI, implying that statistically the bulk
of Ly$\alpha$ photons expected to be produced by massive stars in the
star-forming region hosting the GRB will be surrounded by such opaque lines of
sight. We interpret our results in the context of more sophisticated models and
of different dominant Ly$\alpha$ emitting regions. We also compare LAE-LGRBs to
LAE Lyman continuum (LyC) leakers in the literature in terms of properties
identified as possible indirect indicators of LyC leakage. [Abridged]
