Exquisite Archangel vs. decking out: If you would attempt to draw a card while there aren’t enough cards left in your library, the Angel will prevent you from losing and exile itself, but you still won’t have any cards left in your library. You’ll probably have one turn to win the game, so use it wisely!
Exquisite Archangel vs. Triskaidekaphobia: If you’re at 13 life when either ability resolves, the Angel will prevent you from losing, exile itself, and put you back up to your starting life total (hopefully giving you a buffer before you go back to 13 life), but the ability will continue resolving, and you’ll gain or lose 1 life as appropriate. If both players have 13 life, however, good news for you–the game would normally be a draw, but the Angel replaces you losing with not-losing, while your opponent loses. Therefore, you win!
Exquisite Archangel vs. Mechanized Production: Unfortunately, the Angel won’t have any effect here, as “win the game” effects are separate from game rules and effects that say you “lose the game.” The only exception is in multiplayer games, where “win the game” instead means “all your opponents lose the game”; therefore, you will be protected by the Angel, but your other opponents will still lose, probably reducing the game to a 1v1.
Exquisite Archangel and Platinum Angel: It’s highly unlikely, but possible you’ll be able to pull this one off in Limited! If you do manage to get them both in your sealed or draft pool, and both onto the battlefield at the same time, once again the Archangel won’t have any effect. Platinum Angel straight-up prevents you from losing, so the Archangel won’t have anything to replace.
Exquisite Archangel in Two-Headed Giant: You and your teammate win and lose as a team, so anything that would cause your teammate to lose will also be replaced by the Archangel. For purposes of effects that care about lifegain (or loss, if your life total was somehow higher), though, only you will gain the life.