  In this paper we study the theory of knotoids and braidoids and the theory of
pseudo knotoids and pseudo braidoids on the torus T. In particular, we
introduce the notion of {\it mixed knotoids} in $S^2$, that generalize the
notion of mixed links in $S^3$, and we present an isotopy theorem for mixed
knotoids. We then generalize the Kauffman bracket polynomial, $<;>$, for mixed
knotoids and we present a state sum formula for $<;>$. We also introduce the
notion of {\it mixed pseudo knotoids}, that is, multi-knotoids on two
components with some missing crossing information. More precisely, we present
an isotopy theorem for mixed pseudo knotoids and we extend the Kauffman bracket
polynomial for pseudo mixed knotoids. Finally, we introduce the theories of
{\it mixed braidoids} and {\it mixed pseudo braidoids} as counterpart theories
of mixed knotoids and mixed pseudo knotoids respectively. With the use of the
$L$-moves, that we also introduce here for mixed braidoid equivalence, we
formulate and prove the analogue of the Alexander and the Markov theorems for
mixed knotoids. We also formulate and prove the analogue of the Alexander
theorem for mixed pseudo knotoids.
