The particle has uniform speed, but not uniform velocity, since direction is constantly changing (and velocity is the vector, speed is the scalar).
Kinetic energy is a scalar. Momentum is a vector. (Physics isn't very rigorous about the following) Technically, in the kinetic energy formula, we are using the scalar speed, not velocity. In the momentum formula, we are using the vector velocity (multiplying it by a scalar, mass).
Effectively momentum retains direction, so since direction changes, momentum changes but kinetic energy does not.
Regarding your second question, what is rotating and around what?