Days ahead of you; people have been complaining about this on Twitter all week.
Nope, Weakness Policy triggers after Burning Jealousy's secondary effect.
DragonWhale also mentioned elsewhere that anecdotally, Quick Draw felt way more trigger happy than Quick Claw, which I also got a sense of before I switched to using an Own Tempo Slowbro. So I took the Quick Draw Slowbro out again, clicked attacking moves 220 times and got 68 activations, a breakdown of which is below (1 = activated, 0 = not).
01000000101110100110
00100101000110010101
01000010000000001100
11010000000001101000
00000100000000110011
10101111000000000111
00100000001011010000
01101010000011100000
00001101010000001010
01100001110001100000
00000000000011000100
It also felt very streaky, but that's RNG for you. 68/220 ≈ 30.91%
I'm pretty confident now that
Quick Draw has a 30% chance to activate. Feel free to test for yourself and add your results to these!