Nicol Bolas, Dragon-God

In a Two-Headed Giant game, if one player on a team loses the game, the entire team loses the game, even if the other player controls a legendary creature or planeswalker.

If any abilities trigger as you draw a card while resolving Nicol Bolas's first loyalty ability, those abilities don't resolve until after your opponents have exiled a card or permanent.

While Nicol Bolas may end up with an astonishing number of loyalty abilities, you can still activate only one loyalty ability of Nicol Bolas, Dragon-God during each of your turns.

After you draw a card as Nicol Bolas's first loyalty ability resolves, the next opponent in turn order (or, if it's somehow an opponent's turn, that opponent) chooses a permanent they control or chooses a card in hand without revealing that card. Then each other opponent in turn order does the same, knowing the choices made before them. Finally, all of the chosen permanents and cards are exiled simultaneously.

Nicol Bolas doesn't remove loyalty abilities from the other planeswalkers.

If an ability of a planeswalker exiles cards and another loyalty ability of that planeswalker references the exiled cards, those abilities are linked. If Nicol Bolas gains both abilities from the same planeswalker, the instances he has are similarly linked. Cards exiled by any other abilities Nicol Bolas has (such as his own first loyalty ability) won't be seen by those linked abilities.