  Unified graded differential algebra, generated by $\kappa$-Minkowski
noncommutative (NC) coordinates, Lorentz generators and anticommuting
one-forms, is constructed. It is compatible with $\kappa$-Poincar\'e-Hopf
algebra. For time- and space-like deformations, the super-Jacobi identities are
not satisfied. By introducing additional generator, interpreted as exterior
derivative, we find a new unique algebra that satisfies all super-Jacobi
identities. It is universal and valid for all type of deformations (time-,
space-, and light-like). For time-like deformations this algebra coincides with
the one in \cite{sitarz}. Different realizations of our algebra in terms of
super-Heisenberg algebra are presented. For light-like deformations we get 4D
bicovariant calculus, with $\kappa$-Poincar\'e-Hopf algebra and present the
corresponding twist, which is written in a new covariant way, using Poincar\'e
generators only. In the time- and space-like case this twist leads to
$\kappa$-Snyder space. Our results might lead to applications in NC quantum
field theories (especially electrodynamics and gauge theories), quantum gravity
models, and Planck scale physics.
