I guess the specific question also, does Chandra's ability considered to be red colored?
Correct, protection from red would cause the ability to fizzle and your creature to survive.
A creature with protection from red can't be targeted by abilities from a red source:
702.16b A permanent or player with protection can’t be targeted by spells with the stated quality and can’t be targeted by abilities from a source with the stated quality.
When someone activates a targeted ability, the player has to check twice whether the target is legal: once on activation, and again on resolution. If all targets of that ability become illegal between the checks, the ability "fizzles", i.e. it is removed from the stack without having an effect.
- Casting Spells
601.2c The player announces their choice of an appropriate object or player for each target the spell requires. [..]
The rules for casting spells and activating abilities are the same as far as targeting is concerned.
- Resolving Spells and Abilities
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. [..] If all its targets, for every instance of the word “target,” are now illegal, the spell or ability doesn’t resolve. It’s removed from the stack and, if it’s a spell, put into its owner’s graveyard. [..]
However, protection is defined by a relatively narrow set of rules, which are often communicated using the mnemonic acronym DEBT. The object with protection cannot be:
Damaged by sources with the specified quality. (All such damage is prevented.)
Targeted by spells with the specified quality, or by abilities from sources of that quality.
Source, in the context of Magic rules, can mean one of two things: source of an ability and source of damage.
So protection does not look at the color of the ability, it looks at the color of the source of the ability, which in this case is Chandra.
The wording of the ability is
Chandra, Awakened Inferno deals X damage to target creature or planeswalker.
Since it's Chandra that's dealing the damage, and Chandra is red, protection applies.