The "Khans" ability happens after you draw the card you normally draw during your draw step. That card and the additional card will be in your hand when you have to discard a card.

The "Dragons" ability adds per spell, not per target. It won't cause a spell to cost more than more to cast, even if that spell targets you and a permanent you control or more than one permanent you control.

Each Siege will have one of the two listed abilities, depending on your choice as it enters the battlefield.

The words "Khans" and "Dragons" are anchor words, connecting your choice to the appropriate ability. Anchor words are a new rules concept. "[Anchor word] — [Ability]" means "As long as you chose [anchor word] as this permanent entered the battlefield, this permanent has [ability]." Notably, the anchor word "Dragons" has no connection to the creature type Dragon.

Each of the last two abilities is linked to the first ability. They each refer only to the choice made as a result of the first ability. If a permanent enters the battlefield as a copy of one of the Sieges, its controller will make a new choice for that Siege. Which ability the copy has won't depend on the choice made for the original permanent.