All of the statements are correct:

I. Sedimentary rocks are formed at Earth`s surface by the hydrological system: Sedimentary rocks are primarily formed through the processes of weathering, erosion, transportation, deposition, and lithification, which occur at or near the Earth`s surface. These processes involve the action of water, wind, ice, and gravity.

II. The formation of sedimentary rocks involves the weathering of pre-existing rocks: Sedimentary rocks are derived from the breakdown and weathering of pre-existing rocks. Over time, the exposed rocks are subjected to physical and chemical weathering processes, which break them down into sediments.

III. Sedimentary rocks contain fossils: Sedimentary rocks are often rich in fossils. Fossils are the remains or traces of organisms preserved within the layers of sedimentary rocks. The process of fossilization occurs when the remains or traces of plants and animals are buried and preserved within sedimentary layers.

IV. Sedimentary rocks typically occur in layers: Sedimentary rocks are characterized by their layered structure, known as bedding or stratification. This layering occurs due to the deposition of sediments over time. Each layer represents a distinct period of deposition, which can provide valuable information about Earth`s history and the environmental conditions at the time of formation.

Therefore, all of the statements (I, II, III, and IV) are correct.