Invisible Shields and Rock Skin: At the end of your turn, choose one: Energy Shield/Deflection, Stealth, or Toughness. Super Skrull can use the chosen power until your next turn.

Rock Fists and Elastic Arms: Super Skrull may use Close Combat Expert. When using Close Combat Expert, Super Skrull may target an opposing figure within 3 squares and line of fire.

If my Super Skrull could suddenly use Invisible Shields and Rock Skin during my opponent's turn (maybe it had been Outwitted and the Outwit character pushed off of Outwit), can I now choose which power I can use?
No. You can only choose at the end of your turn. If it isn't the end of your turn, then you just can't choose.

That said, if you had chosen at the end of your turn and then lost the power, then the power came back (like in your Outwit example), whatever you had chosen would return.

If I use Close Combat Expert and target a character that is not adjacent with Rock Fists and Elastic Arms, does the opposing character get to use Energy Shield/Deflection or hindering terrain modifiers?
No. Even though you must be able to draw line of fire to the target (so Stealth might be an issue), the attack itself is still a close combat attack. This means no Energy Shield/Deflection, no hindering terrain modifier to defense value, and you can even make the attack if there is another opposing character adjacent to Super Skrull. But it also means that your target could use Combat Reflexes and other close combat related effects.