(a) Yes, a body at rest has no momentum i.e., p=0 and as suchits kinetic energy (= p^2 /2 m)=0. But it can have potential energy (U) by virtue of its position or configuration. Thus, its total energy, E = K + U = U and is not zero, e.g., a stone lying on a roof or a wound up spring of a clock.
(b) Yes, if E = 0, K + U = 0 i.e., either K = U = 0 or K = - U. Thus, when E = 0, the body can possess KE and on account of that possesses momentum. An electron in an atom has momentum even when its total energy is negative. But note that a. body, which has momentum, must be in motion and consequently possesses kinetic energy.