Well if the Shield can't be destroyed,doesn't it negate Nanobots or Kitty Pride ability to take it out of the game???

No, because you aren't destroying it, just removing it from the game. Very different.

This really makes Kitty annoying when using her against the new captain.

Also, in regards to removing objects from the game does this also include immobile ones such as Mjolner and the Gem? How about removing them when they have been picked up? How about when they have not been picked up or have been dropped?

Just like the Shield, Mjolnir and the Gem can't be destroyed, but they can be removed by Kitty if they're in the same square or adjacent to her.

However, Mjolnir and the Gem aren't held, they're just placed on the character card, so Kitty wouldn't be able to remove them if an adjacent character has picked them up.