The key to completing Part 1 is to use the hierarchy of dilution discussed in class. It starts with turbidites (which overwhelm everything else) and goes on to radiolarian oozes, carbonate oozes, and finally red clays (which are deposited only where nothing else is deposited).
With that in mind, you would put in areas of turbidites first around the continents, then put diatom and radiolarian oozes in the remaining areas of the appropriate latitudinal belts, then put carbonate oozes in the remaining areas where abyssal depths are appropriate, and finally put red clays in the remaining abyssal areas.
Evaporites and shallow carbonates aren't deposited in the open abyssal ocean. Evaporites are deposited in restricted seas in the Horse Latitudes. Shallow carbonates are a likely candidate in shallow seas not in the Horse Latitudes.
The key to Part 2 is to figure out the path along which each numbered piece of crust has moved, and then to deduce the sediments that it would have accumulated along the path. Figuring out the path requires you to think about plate tectonics and where seafloor originates.