Is a global water cycle possible without an atmosphere? Without the lithosphere? How do they participate in the water cycle?

Not possible, since all shells are interconnected. If there was no atmosphere, then there would be no fresh water on Earth, since fresh water evaporates in the form of steam, precipitation is formed. Through rocks, water seeps, forming underground water, which then flows into rivers and lakes.
Water from the surface of the oceans evaporates and forms water vapor, which is carried by the wind throughout the planet. Once in the upper atmosphere, water vapor condenses and may precipitate as snow or rain. Precipitation occurs both above the surface of the waters and the lithosphere, where they soak into the soil layers, form rivers and underground ground currents, which transport them back to the waters of the seas and oceans. Therefore, the world water cycle without the atmosphere and lithosphere is not possible.