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

Nicol Bolas doesn't gain any static or triggered abilities of other planeswalkers or any activated abilities they may have that aren't loyalty abilities.

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.

If a loyalty ability of a planeswalker references the card it's printed on by name, treat Nicol Bolas's instance of that ability as though it referenced Nicol Bolas, Dragon-God by name instead. For instance, if Ral Zarek is on the battlefield, Nicol Bolas deals the damage if you activate the second ability he gained from Ral.

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.

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.

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.

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.