Yes, Starship needs legs to land on the Moon or Mars. They will be lighter than legs for Earth landings, as Moon/Mars have lower gravity.
Not sure about the amount of fuel needed to land, as there is much less atmosphere on Mars, and none on the Moon. I would still guess that it needs less fuel there as well.
An atmosphere reduces the amount of fuel needed to land, because you can use aerobraking to slow down rather than carry fuel to do it. See the Apollo return capsule, which landed without any rockets, only parachutes and a heat shield.
Not sure about the amount of fuel needed to land, as there is much less atmosphere on Mars, and none on the Moon. I would still guess that it needs less fuel there as well.