Hello, just by looking at the Japanese tooltip for Dripping Blades we get that it only affects Physical skills, the wording "Action" is wrong in this case as it still affects Fuma Shuriken but not Raiton.

As for skillspeed, first of all the tooltip states that only weaponskill damage over time is increased with skillspeed, but this does also increase the damage dealt by abilities, Dark Knight's Salted Earth and Ninja's Doton being two examples.

Secondly, it completely lacks any information of it's effect on Auto Attacks, where it should also be added that it increases the potency of the auto attack just like how it works with damage over time.