If you activate Chandra Ablaze's first ability, you don't discard a card until the ability resolves. You may activate the ability even if your hand is empty. You choose a target as you activate the ability even if you have no red cards in hand at that time.

As the first ability resolves, nothing happens if your hand is empty. But if you have any cards in hand, you must discard one. If you discard a nonred card, Chandra doesn't deal any damage.

If the creature targeted by the first ability is an illegal target by the time it resolves, the entire ability doesn't resolve. You won't discard a card.

You may activate Chandra's second ability even if your hand is empty. As it resolves, a player whose hand is empty simply draws three cards.

You cast red instant cards and red sorcery cards from your graveyard as part of the resolution of Chandra Ablaze's third ability. You don't choose which ones to cast until you're actually doing so as the ability resolves. You cast only the ones you want to, and you may cast them in any order. Timing restrictions based on the card's type (if it's a sorcery) are ignored. Other restrictions are not (such as "Cast [this card] only during combat"). Each card you cast this way is put on the stack, then the ability finishes resolving. Those spells will then resolve as normal, one at a time, in the opposite order that they were put on the stack. They'll go back to the graveyard as they resolve.

If you cast a card "without paying its mana cost," you can't pay any alternative costs. You can pay additional costs, such as kicker costs.