What happens to an investigator who arrives at a gate already explored by another?
Yes, you're correct about both.

Until the Arkham encounter phase, it doesn't really matter that there's a gate there. So you can do whatever you like during movement, including exchanging items.

When you arrive at the Arkham encounter phase, each player in a location with a gate will be drawn in if they haven't been through the gate before. So depending on turn order, as you say, either the player who's been through can close/seal the gate first, or the other player will be drawn in.