  We consider a model of a Josephson junction of two topological
superconducting wires mediated by an interacting quantum dot. An additional
normal electrode coupled to the dot from the top allows to probe its density of
states. The Majorana states adjacent to the dot hybridize across the junction
and from a bound state in the dot. The dot is subjected to the effective
magnetic field arising from the superposition of the fields driving each wire
into topological states, which, dependent on the angle between the fields,
introduces variable Zeeman splitting of the dot active level. We show that
electron interactions in the dot diminish the characteristic for Majoranas zero
bias peak arising in the transverse conductance through the dot and introduce
an overall asymmetry of the conductance. They also renormalize the
hybridization between the end-state Majoranas in shorter wires. The Majorana
spin polarization is determined by the effective magnetic field in the dot.
Phase-biased Josephson current exhibits spin polarization in thermal
equilibrium, which possesses characteristic $4\pi$ periodicity, and its sign
can be switched when an unpaired Majorana state is present in the junction. We
also observe spin-dependent Majorana state "leaking", which can be controlled
by the position of the dot level in energy scale.
