It's about time, too.
I hate overtext but love irony.
Playing solo. Since card 18 instructs to place Star tokens equal to number of survivors, I place one star token on the highest value settlement. I happen to be on that space and kill the Star enemy, which is the last Star token on the board. So I stage 024. Since no Stars are on the board, the Courser moves at the end of each round closer to Railroad. Is this movement at the end of the round considered the "leading" that is described in the objective?
Yup. You don't need to be there yourself, he'll find his own way there.
For card 132, does this card stay on the board throughout the whole game, since there is no Trash directive on it? It appears you can use it immediately if you have an action left, since you are already there. If it does stay on, then the only time you can use it if it you move back to that quest marker?
You can use it as often as you want. Twice per turn if you like. Go nuts, drive them robots around. Feel the power. But then eventually remember to go get some influence or something.