Having gathered up to 800 Mythril Ore, equal numbers on other items, I can confirm that HQ is around 15-20% and it's like that all the time. Proc for Mooch is similar. For Gathering, it's 15-20% of hits though so if you are getting nothing 40% of the time, you get 15% of the 60 times you hit.

Crafting is different. If I can get 50% or 75% or 88% with all Normal condition and do it over 100 Synthesis, my true HQ will come up half on those synthesis. The number fusges based on Good quality but the true value of my crafting percentage is almost always half. It doesn't matter what the bar says.