If you pick up a Spirit Monster, you cannot Special Summon it and it is sent to the Graveyard.
If you pick up a Guardian and don’t have the corresponding Equip Spell Card on the field, you cannot Special Summon the Guardian and it is sent to the Graveyard. If you have the corresponding Equip Spell Card, the Guardian is Special Summoned.
If you go through your whole Deck without finding a card to pick up, “Monster Gate”’s effect is negated and the cards you picked up are shuffled back into the Deck.
Your opponent can chain “Royal Oppression”’s effect to negate “Monster Gate”’s effect.
Cards picked up that cannot be Normal Summoned are considered “sent to the Graveyard”, not “discarded”.
If you Special Summon “Dark Magician of Chaos” with “Monster Gate”, the effect of “Dark Magician of Chaos” activates, but it activates after “Monster Gate” has fully resolved and sent other picked up cards to the Graveyard, so you can use “Dark Magician of Chaos”’ effect to retrieve a Spell Card that was picked up and sent to the Graveyard by “Monster Gate” in the process of searching for the “Dark Magician of Chaos”.