I've tried to answer all your questions. I probably included some extra stuff.
Basic attacks are a "regular swing" in a battle. A hero performs a basic attack when they have no skills ready to use. Green/Blue/Purple/Orange skill readiness is not information that is not directly communicated, but rather is something that must be observed and deduced from a hero's actions in a fight.
For instance, at the start of battle Corgi normally performs a basic attack (splash melee to enemy front line), then her green skill, then her blue skill, then some basic attacks, and the cycle repeats. However, if Corgi is blinded before she can perform her green skill, she will do nothing while blinded until her blue skill is ready, which would seem to be a magic-based skill that can be used while blinded. Once recovered from blind, Corgi will immediately use her white skill if she can, and then will likely use her green skill next. Her normal attack pattern may be altered for the rest of the fight.
Skill power scaling
Example Hero: Spikey Dragon
Spikey's basic damage scales with his primary stat, Agility.
Spikey's white skill scales with skill power.
Spikey's blue skill scales with basic damage.
Spikey's purple skill scales with skill level only.
On their own, they shouldn't affect damage or anything else. An attack or skill just "is" its damage subtype(s). A slashing attack should just be a slashing attack with no special bonus, penalty, or behavior. Other games might call Dragonsoul's damage subtypes "properties". On a conceptual level, I've seen these referred to as "tags".
EDIT: Let's say, for example, a basic attack or skill has the Fire and Necrotic subtypes and deals 5000 damage. 100% of that damage is Fire and 100% is also Necrotic, but it still only deals 5000 total damage. Any Fire or Necrotic damage bonuses or penalties will modify that 5000 damage accordingly, so a 10% Fire damage bonus will make the attack deal 5500 Fire (and Necrotic) damage. That's my understanding, anyway.
When damage types might matter
Certain hero skills
Boss pit requirements
How to use this information to your advantage?
When determining which gear will have the largest impact on each hero, and then to prioritize collecting scraps for those items first and how much to invest in enchantments for all gear. For example, Roller Warrior at Orange (zero) needs Photo Bomb. Photo Bomb gives skill power and intellect, but none of Roller Warrior's skill variables scale off skill power and intellect is not her primary stat. Thus, Roller Warrior only benefits a very small amount of magic resist from Photo Bomb's intellect increase, so Photo Bomb is a very low priority for Roller Warrior and enchanting it is almost a complete waste. Eventually, though, you will want to max enchant suboptimal gear anyway. Awareness of this enables better resource management.
Next is determining when to promote. Basically, you should look at a promotable hero's current enchantment bonuses (refer to the green numbers on Spikey's stat page above) and compare that against what you would gain from gear that you could immediately equip that hero with, both core stats and gains from enchanting, if promoted. Since promoting a hero removes their enchantment bonuses, you want to at least somewhat break even on stat gains if promoted and enchanted. To that point, I haven't promoted most of my O+1 heroes even though their gear sets are complete because their O+1 gear is very good for most of them--thus, I've max enchanted most of that gear--whereas their O+2 gear provides markedly worse gains in most cases. Another way to look at that is the power gap between O+1 and O+2 heroes isn't very influential. Again, awareness of this enables better resource management.
I broached that thinking in an earlier post here.
Lastly, there's runes. I have no personal experience with runes, but my impression is that runes can increase almost any stat, including damage belonging to specific subtypes, like Fire on Kaiju. Awareness of which stats influence which heroes and to what degrees should help guide rune selection on a per-hero basis, but I imagine some superior rune configurations are not always the most obvious.
Increasing basic damage should always improve a hero's basic attack damage. Skill variables that scale with basic damage (i.e., have brown-colored text) will also improve, but that scaling is different for every power (or should be thought of as such).
How does knowing that help with gear, enchantment, and promotion decisions?
Obviously, a hero that relies heavily on scaling from their basic damage should have greater importance attached to added basic damage gear and enchantments for that gear, as well as gear adding to their primary stat. Furthermore, bonus crit damage (e.g., from Roller Warrior's orange skill), crit rating, and armor/magic penetration indirectly improve the usefulness of high basic damage.
How does knowing that help with hero selection?
I don't suppose it does. Aquatic Man can reduce an enemy's basic damage and Master Thief can steal an enemy's basic damage, so some knowledge of basic damage could help when fighting with or against those. Knowing whether a hero's basic attack can hit multiple targets (i.e., "splash damage") or only a single target can also influence decision-making for basic damage, but that information is not adequately communicated outside of battle. Also, having a rough estimation of the armor and magic resist toughness of enemies in a defending line can influence your offensive hero selection, which in some cases means considering your heroes' basic damage.
You can try looking up a hero on the wiki for its skills' scaling formulae. Presumably, those equations are correct. For example:
This will probably be my one long, helpful post for a month or two.