I have the exact problems with those 3 skills.
Can confirm serpent strike was not doing any damage as soon as I change to a different weapon from what I initially had equipped.
I made a new primalist today intending to use tempest strike as one of its skills. I decided otherwise after discovering its visual is bugged atm . Indeed, purple/pink squares.
Some update.
I have changed my weapon back to spear, and the skill is working fine again.
Just noticed the skill does require a spear.
As long as you are using a spear, the skill works.