Astrophysics and Aerospace Engineering degree here. You talk about energy and momentum, so I’ll cover both. I’m also going to stay with Newtonian physics.
Assume the energy to start was the potential energy at the top of the slide. As you started to slide down, you gain equal kinetic energy from lost potential energy (minus friction). When you stop with your hands, you convert the gained kinetic energy to friction losses (heat and sound). You basically converted your distance from Earth’s gravity well for heat and sound.
Momentum is conserved with two momentums in opposite directions. When you climbed up the slide, you pushed off from the Earth’s mass. Both move, but you move far more because your mass is far smaller. When you slide down, you release a normal force that prevented the Earth and you from moving closer to each other from gravity. Again, you move far more in relation to the Earth. When you stop with your hands, you re-apply the normal force. Then, both you and Earth stop moving in relation to each other. People are saying that you slightly move Earth when you stop, but you actually stop slightly moving Earth when you stop.
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