(i) No.
Consider the mechanical energy of a particle under gravity. The total mechanical energy = K.E. + P.E.
K.E. is always positive because either v is positive or negative, v² will always be positive and so will ½mv² = K.E. but P.E. = mgh may be positive or negative because h is positive or negative depending upon its position with respect to the reference line. Suppose we take ground as the reference line. If the particle falls into a dry well, at some depth below ground level the magnitude of negative P.E. may be equal to K.E. thus making the total mechanical energy of the particle zero. But due to its velocity, the linear momentum is not zero.
(ii) No. It is not necessarily nonzero. Suppose the particle is on the ground at rest. K.E. = 0, and P.E. = 0. Total mechanical energy = 0. Since velocity is zero, the linear momentum is also zero.
