Say my opponent attempts to Charge my Kingdom Come figure, but I make my KC roll.

Is my opponent forced to finish off the Charge, using the object in hand, if there are other viable targets available, such as figures or terrain?

Or can the figure meekly finish out the remainder of its Charge movement without being forced to finish off the close combat attack portion of the move?

I'm going to try to answer this one for you. Since you are talking about a charge, this seems simple to me. Since the first half of a charge is considered movement (power action) then before you even started to do the charge, KC dice roll would have to be made. This is from the DC team abilities page on Wizkids' website:

KINGDOM COME: Whenever an opposing character given a move action, power action, or free action attempts to move from a square that is not adjacent to the KINGDOM COME team member to a square adjacent to the KINGDOM COME team member, roll one six-sided die before moving the opposing character, even if it ignores the effects of characters on movement. On a result of 1 or 2, the opposing character cannot move to any square adjacent to the KINGDOM COME team member that turn.
Characters with this team symbol and characters suffering knock back ignore this team ability.

So according to this, if the roll is successful then you wouldn't even be able to charge a KC character as it would have to end in square adjacent to that character. As for charging another character in range, I believe it's possible (I may be wrong) as long as it doesn't bring the charger adjacent to the KC figure that made the successful roll of the die. Hope that helps.

Pretty sure it's the same as if you went to charge a Skrull/Shape Changer. Once the role succeeds, your action is null and void and you must do something else.

They do not work the same. (And your description isn't exactly how Skrull interacts with Charge.)

With KC, you are making the roll during the movement. If the Charger cannot move adjacent, it can continue its movement to another target or none at all.

As for the Skrull thing, the Skrull/SC roll happens after the movement is complete and before the attack. If the roll causes you to not be able to attack, the action isn't null and void, as it still counts towards your actions. You also token the figure.