A spell or ability that counters spells can still target Chandra. When that spell or ability resolves, Chandra won't be countered, but any additional effects of that spell or ability will still happen.

If Chandra's last ability doesn't deal damage (most likely because the damage has been prevented in some way), the target won't be exiled if it would die.

Emblems aren't permanents and can't be exiled or destroyed. Once you have an emblem, there's no way to remove it.

A player may have more than one of Chandra's emblems. Each one's ability triggers separately.

In a Two-Headed Giant game, Chandra's first ability grants an emblem to each opponent. The opposing team will be dealt 1 damage twice during their upkeep.

The ability of Chandra's emblem triggers at the beginning of the upkeep of the player who gets the emblem, not the upkeep of Chandra's controller.

The replacement effect of Chandra's last ability will exile the target creature or planeswalker if it would die this turn for any reason, not just due to lethal damage.

If you leave a multiplayer game, your opponents keep any emblems Chandra gave them. Each opponent owns the emblem Chandra gives them. In multiplayer, your opponents continue to burn after you and Chandra leave the game.