Party member tantrums seem to be fairly "harmless" (in that they normally just murder someone, often some goblin nobody cares about) and I don't think a character you are currently controlling can tantrum at all.
However, fullfilled needs DO matter quite a lot, and those stats affect them. First of all Love and Lust are both VERY VERY VERY BAD in adventure mode (despite lust being a good thing in fort mode. I don't even think fulfilling the "make romance" need in adventure mode is even possible, and not just in a "you can't actually start a romance without hacking" way (and you can't), but even if you hack someone to be your lover it doesn't seem to fulfill the need.
Honestly most positive emotions are really bad for adventurers. Spending time with friends and family is basically impossible. Helping people is supposedly possible, but for the life of me I can't get the reunions to trigger and even if I could its easily the most random and time consuming need to fill.
Ironically being a horrible person is the shortest path to happiness. Adventurers who like fighting, arguing, causing trouble, and being extravagant are really easy to keep fulfilled and happy.
As for just minmaxing stress, maxing out joy propensity or minimizing stress vulnerability are enough to render someone virtually immune to stress by themselves (stress vuln of 0 just won't get bad thoughts at all, joy propensity of 100 can get beaten and arrested 6 times in one year and still be the happiest person in your fort by a massive margin). putting both together is enough to make someone who could be happy even when caught in the rain, retching on miasma, surrounded by the corpses of their family.