Yes, a Pokémon can find a box and a key. However, that will almost
never happen if you aren't using very long missions, and even then,
it's quite rare.
No. Longer missions give more items (and, by extension, more
chances to get items), but as far as I'm aware, any item can be
found on any mission if you're lucky.
They have existed since Keys were first added; they just aren't
available yet. Use the [item] code for proof that they are real -
the images in the first post are all directly from the site itself.
To open a Box you have to go to your Item Bag, to the tab Special,
and there it will say: "It looks like you got your hands on some
mystery boxes. Click here to open them." If you click where it says
"Click here", you'll be able to open the Boxes, if you have the
respective keys :3
You push the dark blue key carefully into the hole, turn it twice
to the left and... cccrrk!!
The mystery box opens and you look curious in it...
4x Fairy Gem found!
The key breaks and becomes useless.