Chandra's first ability triggers if she loses loyalty from being dealt damage, from her last loyalty ability being activated, or because an effect removes loyalty counters from her. If an effect simply causes Chandra to leave the battlefield, no counters were removed and so her ability doesn't trigger.

If all of Chandra's loyalty counters are removed, her first ability has her deal damage even though she's left the battlefield.

Chandra's loyalty abilities don't change when you can play the exiled cards. For example, if you exile a sorcery card, you can cast it only during your main phase when the stack is empty. If you exile a land card, you can play it only during your main phase and only if you have an available land play remaining.

Casting an exiled card causes it to leave exile. You can't cast it multiple times.

If multiple creatures deal combat damage to Chandra at the same time, those counters are all removed at once, and her first ability triggers only once.