If momentum is conserved in a collision, is kinetic energy conserved too?

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These are two different conservation laws with two different conditions. Linear momentum of a system is conserved whenever the net external force on it is zero (or negligible during the brief impact). The forces the colliding bodies exert on each other are internal third-law pairs, so they cancel for the system. That holds for every such collision, whether the bodies bounce apart or stick together.

Kinetic energy is conserved only in perfectly elastic collisions. In most real collisions some kinetic energy turns into heat, sound and permanent deformation. Total energy is always conserved, but kinetic energy alone usually is not.

Clay balls that stick together make a perfectly inelastic collision, which loses the most kinetic energy that momentum conservation allows. Example: a 1 kg ball at 4 m/s hits an identical ball at rest and they stick. Momentum: 1 × 4 = 2v, so v = 2 m/s. Kinetic energy before = ½ × 1 × 4² = 8 J; after = ½ × 2 × 2² = 4 J. Momentum is unchanged at 4 kg·m/s, but half the kinetic energy is gone.

For problems: write the momentum equation first. Use kinetic-energy conservation only if the question says the collision is elastic. For collisions in between, the coefficient of restitution e (0 ≤ e ≤ 1) tells you how elastic the collision is.