No, kinetic energy does not depend on the direction of motion involved. Kinetic energy can never be negative. Yes, the value of kinetic energy depends on the reference frame of the observer.
Kinetic energy (K) of a particle is defined as the,
K = ½ mv2
Where m is the mass of the particle and v is the velocity of the particle.
Since kinetic energy is a scalar quantity, therefore there is no direction associated with kinetic energy and it has no components. Thus kinetic energy does not depend on the direction of motion involved.
Since, mass of the body (m) is always a positive quantity and the term v2 in the kinetic energy is always positive, therefore, kinetic energy of a body is always a positive quantity. Therefore kinetic energy can never be negative.
Newton’s laws are valid only in inertial frames of reference. If we find Newton’s second law to hold in one frame of reference, then it holds in all inertial frames. If two observers in different inertial frames move at constant velocity v relative to one another and observe the same experiment, they measure identical values for the forces, masses, and accelerations, and so they agree completely in their analysis using Newton’s second law.
In the Newton’s second law, observers in different inertial frames will agree on the results of applying the work energy theorem. However, unlike forces and accelerations, displacements and velocities measured by observers in different inertial frames will in general be different, and so they will deduce different values for the work and kinetic energies in the experiment. Therefore the value of kinetic energy depends on the reference frame of the observer.