Intact breeding cats often spray. When neutered/ spayed at an appropriate age (5 months or earlier as recommended by your vet is optimum) Savannahs are not known to spray.
A well socialized, happily acclimated Savannah will use its litter box religiously! Like most cats, some Savannahs can absolutely require their boxes be extremely clean, and you must make sure there are plenty of litter boxes for the number of cats in your house. The general rule is one litter box per cat and one extra. Additionally, Savannahs can grow to larger-than-normal sizes, so you may need the jumbo-sized litter pans. Some people find that plastic tote boxes make excellent litter boxes, especially with those cats that like to dig and fling litter about.
Any cat litter is acceptable.