  We address the interplay of few lattice trapped bosons interacting with an
impurity atom in a box potential. For the ground state, a classification is
performed based on the fidelity allowing to quantify the susceptibility of the
composite system to structural changes due to the intercomponent coupling. We
analyze the overall response at the many-body level and contrast it to the
single-particle level. By inspecting different entropy measures we capture the
degree of entanglement and intraspecies correlations for a wide range of intra-
and intercomponent interactions and lattice depths. We also spatially resolve
the imprint of the entanglement on the one- and two-body density distributions
showcasing that it accelerates the phase separation process or acts against
spatial localization for repulsive and attractive intercomponent interactions
respectively. The many-body effects on the tunneling dynamics of the individual
components, resulting from their counterflow, are also discussed. The tunneling
period of the impurity is very sensitive to the value of the impurity-medium
coupling due to its effective dressing by the few-body medium. Our work
provides implications for engineering localized structures in correlated
impurity settings using species selective optical potentials.
