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Yes. Kinetic energy is a scalar (non-directional) and momentum is a vector. That means if you have two or more objects in motion their kinetic energies always add. But their momentums must be combined using vector addition and you might get zero. The simple case is two identical masses moving with equal but opposite velocities. Their total momentum is zero because their directions of motion are opposite.

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Yes,

Take two objects of the same mass, A and B, whose velocities, VA and vb, are opposite

Ke = .5(m)(VA)^2 + .5(m)(vb)^2 =

Plug in some numbers, m = 1 kg, VA = 10 m/s, vb = -10 m/s

.5(1)(10)^2 + .5(1)(-10)^2 = 50 + 50 = 100 -- a positive number

momentum can be found using p = mva + mvb

so p = (1)(10) + (1)(-10) = 10 + -10 = 0

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