Lightning Strike mechanics and support gems

Lightning strike is an attack, not a spell. Not spell specific modifiers will have any effect on it.

The initial part of the skill is hitting with a melee weapon - this is melee attack damage.

The projectiles also deal your weapon damage, but reduced, as the skill has a reduction applied to projectile damage, which you can see on the gem.

There are two specific rules that modify the projectile's behaviour - If the initial melee attack hit, the projectiles can't 'miss' due to accuracy/evasion (they can of course still 'miss' if you send the projectiles in a direction where there's nothing to hit)
In addition, if the melee attack hits something, that thing can't be hit by the projectiles - otherwise it always would be since they are created right on it.