As my current findings, I can't see any significant changes when I use the Gambler Ring while I am on havoc form, it also didn't work on mods like the one with Nebula.
#Are there rings/amulets/relic fragment that increase crit chance of skills?
primal jetty
graceful kiln
Crit Chance is an overall stat, should affect everything. There is a relic fragment for crit chance and crit damage. Gambler ring as you mentioned as well