If an owner sent his slave to make a korban pesach for him, no matter if the slave slaughters a kid or a lamb, the master can use it. If the slave slaughters both and the master is a king that has so much, he does not care which kind of animal he gets, he can eat from the first.

If the slave forgot what his master told him, a kid or a lamb, he should slaughter both with a condition that whichever his owner meant that is the real korban pesach.

If the owner forgot which he said, both animals must get burned but he is not obligated in a pesach sheni.

Pesach 8:3

A father who registered all his children on a korban pesach but wants to give them an incentive to travel to Yerushalayim quickly may say, ‘whoever will get to Yerushalyim first will get the korban pesach and will give his brothers from his animal’.

You can register as many people as you want on an animal of the korban pesach as long as each person gets an olives worth.

You can register or withdraw from the korban pesach until you slaughter it. R’ Shimon says until you sprinkle the blood.