Nyxathid's two abilities are linked: The second one refers only to the player chosen by the first one. If another creature becomes a copy of Nyxathid (due to Mirrorweave, for example), the second ability won't do anything because a player was never chosen as a result of the first ability. This is true even if a different ability allowed a player to be chosen as that creature entered.

Nyxathid tracks the hand size of the chosen player even if that player ceases to be an opponent of its controller. (This could happen if that player gains control of Nyxathid, for example.)

In a multiplayer game, if the chosen player leaves the game, Nyxathid's second ability simply won't do anything anymore.

If a spell or ability causes the chosen player to draw cards and then discard cards, Nyxathid's power and toughness changes accordingly as that spell or ability is resolving. For example, if the chosen player has five cards in hand and a spell causes that player to draw two cards then discard two cards, Nyxathid will start out as 2/2, then become 1/1, 0/0, and finally 2/2. This is because cards are always drawn one at a time, but multiple cards may be discarded at once. Although Nyxathid momentarily had 0 toughness while that spell was resolving, it isn't put into its owner's graveyard as a state-based action because state-based actions aren't checked until the spell has finished resolving.