The emblem’s triggered ability looks for the actual amount of mana spent to cast the spell. If an effect caused you to pay more or less mana for that spell as you cast it, that will be taken into account when determining the value of X. If an effect would counter that spell unless you pay some amount of mana, that mana doesn’t count as mana spent to cast it.

Chandra’s first ability uses the stack and can be responded to, even if no targets were chosen. It isn’t a mana ability.

Chandra’s second ability checks the characteristics of the card you exiled to see if it’s red. If the card is red, but the spell that card becomes somehow isn’t red, you may still cast it. For example, Rowan, Scholar of Sparks, the front face of a modal double-faced card, is red. Will, Scholar of Frost, the back face of the same card, is blue. In exile, that card is red, so you may cast either face if you exile it with this ability.

You must pay all costs for spells cast via Chandra’s last two abilities. For the middle ability, you must also follow all timing restrictions.

Chandra’s last ability, by contrast, only allows you to cast spells that are red when you cast them, no matter what the card in exile is. If Rowan, Scholar of Sparks (a modal double-faced card) is exiled by this ability, you won’t be able to cast its back face, Will, Scholar of Frost, this way because it would be a blue spell.