The full texts of these abilities are "At the beginning of the monarch's end step, that player draws a card" and "Whenever a creature deals combat damage to the monarch, its controller becomes the monarch." (2020-11-10) If the triggered ability that causes the monarch to draw a card goes on the stack and a different player becomes the monarch before that ability resolves, the first player will still draw the card
Thousands would probably not be enough
It's barely even a card game
Since you can always see the top card of your library, Traveling Chocobo lets you plan your turns ahead with the knowledge of what you're drawing if you can't play it
(When this Equipment enters the battlefield, create a 2/2 red Rebel creature token, then attach this to it.) Whenever Blade of Shared Souls becomes attached to a creature, for as long as Blade of Shared Souls remains attached to it, you may have that creature become a copy of another target creature you control