I've found, with a 3.5 year old and a 22 month old, there are a few primary causes of their temper tantrums.
1. exerting their independence - in which case it's our job to bring them back to reality in terms of who makes the rules and runs the roost.
2. being overly tired
3. needing attention but not knowing how to tell us.
4. Desperately needing to get energy out of their systems
My husband and I are very authoritative. Our kids, though we get on the ground and play with them like kids, know we're the parents, it's our house, and we get to make the rules. But, they get to act like kids when appropriate and get treated when they're exceptionally good. Last night, it was 4 peanut M&Ms at the end of dinner for trying all the food on his plate and asking politely to be excused.
Sometimes, when our son's being a complete brat, I'll just stop, look him in the eyes and ask what he wants. Usually, he wants Mommy to spend some special time with him. Sometimes, he needs to run around like crazy and burn off steam. When it's inappropriate (publicly, etc), we have set expectations for behavior, and when they've met those expectations, we make sure they get to blow off their steam when we get home by doing something they want to do.
As for the potty training. Did you praise her when she went on her own? Our son was completely disinterested in getting potty trained. We started forcing the issue after his 3rd birthday. When he would go, we'd make a BIG deal about it. We set daily goals (maybe trying 5 times, then having to go 3 times/day, then moving that to 5 times/day) until he was consistent with it. Now, he's a champ. But, we did have to bribe him in the process. He didn't want stickers or to play "Sink the Cheerios". He wanted Cars. So, my husband spent a bunch of money on them, and when he hit his goals, he'd get one. If he didn't - no car, and we'd try again the next day.
I found this age to be the most challenging because they're smart, capable and trying to be independent.
Good luck. I hear it gets better - we just haven't totally gotten there yet.