fire damage over time multiplier with attack skills, does it helps burning arrow?

I do not quite understand attack skills means even after googling.
Most google mentions are explosive arrows...so does that means burning arrow too, but then, burning arrow does not explodes, so is it a skill? Or is it just an arrow attack.

And, lets say fire dot multipler with attack skills helps burning arrow, does it help with the ignite or something else?

burning arrow is an attack skill, and will benefit from this.
the distinction "with attack skills" on quivers were added as to curb the usage of them with spell builds.

if a skill uses your weapon to do its thing, its an attack skill

It helps the ignites you inflict with Burning Arrow, but not the hit.

If there was an attack skill that inflicted a burning debuff that is not ignite, that debuff would benefit from this stat as well. The old Burning Arrow did that, but it was changed a few patches ago and I do not believe that there is another attack skill with such a component.