I totally hear ya about tantrums - they can be so draining, especially when you're out and about. I usually try to keep my cool and not let the stress show, which is hard when you're also chasing a million other errands!
For pre-emptive measures, I try to make sure my daughter has had enough sleep and eaten well so she's in a good mood, because a hangry toddler is a recipe for disaster. Haha. When she does have the occasional meltdown, which often happens when things don't go her way, I first make sure everyone around is safe - including other kids! - then try to figure out what's really bothering her. Most of the time it's something very minor like a sticker that won't peel off or something, but she's just too upset to speak rationally yet. So I'll help her vocalise her feelings - are you frustrated/sad/upset because your sticker is stuck? Then together we come up with a solution. Sometimes removing her from the situation helps, like going behind a tree to calm down for a moment before we try again.
As for sibling rivalry, since mine are still young, I haven't dealt with it as much yet, but I hear you on the attention-seeking behaviors - my daughter also has some jealousy issues when her baby sister cries, because she thinks I'm giving the baby all my love and attention. So I try to make sure I have little 'mommy-and-me' moments with the older one too, even if it's just a silly dance break or tickle fight, so she feels extra special and loved. That usually helps with the jealousy pangs!