Find a light ball for Pikachu

I had the same trouble finding this option at first, and argued with someone at length that this just wasn't possible. The letters list does seem to be like the old GTS; you can only search for Pokémon you've at least seen.

But if you scroll all the way down past "Z" You'll notice a button called "What Pokemon?" Click on this and it will bring up a text input, where you can manually type in the name of a as-yet-unseen Pokémon and search for it.

It is not possible to shorten a Pokémon's name to search for it. Searching for "Geodude" will return results, but not "Geo". However, do be aware that it does not inform you whether or not the name you typed exists - only that there were (no) results after you search. Also as there are a number of Pokémon that do not normally occur in the Kalos region, searching for them will likely not produce any result until Nintendo releases the Pokémon Bank and this generation's transfer tool.

Also, as event-only Pokémon cannot be traded on the GTS, there are none listed. So if you've manually searched for "Mew" or "Celebi" or any of the other event-distribution Pokémon, it will not yield any results, as there ARE none of those Pokémon for trade on the GTS.

Leveling does not directly affect a Pokemon's EVs (and therefore Super Training.) In prior generations (before V, I believe,) EVs were only calculated upon leveling, so a level 100 Pokemon could not be EV trained. Now, however, EVs are calculated after each battle (and probably between battles, in the case of Super Training.)

However, leveling via battle affects Super Training because it grants EVs based on the Pokemon fought. So leveling via rare candies won't affect super training, but leveling through battle will.

I believe that EXP Share grants EVs to all Pokemon—confirmed by UnderscoreZero in the comments.