When a Character has returned from another world as long as he does not move he can try to close the gate on successive turns, what happens if another Character moves to the same location, can he try to close the gate too or is he sent through the gate to that world.

Strictly from the rules he cannot close the gate. Also if he is still at the gate location by the end of his movement phase he will be sucked through the gate in the encounter phase, unless the gate is closed before that happens.

No. Investigators can only attempt to close gates if they have an explored marker, which is only granted when they return from the other world. And if an investigator goes on a gate he won't be sucked in unless he waits there until the arkham encounter phase. This is important because it means an investigator can go to an open gate with another investigators who has explored the gate, give him an elder sign and leave.