First, let me say that these outbursts are VERY appropriate to the age. My son surprised me with this a few months ago, and at first, it did throw me off. But in case you want reassurance that your kid is right on track, this is from a website I like called "All the Daze" under 'five year old development':
"Can have defiant and aggressive behavior, but still feel indecisive. An emotional outburst may escalate beyond the child’s control."
I agree that helping your child manage their feelings is more progressive than punishing. When my son is getting mad, I gauge my response/actions to what is happening. If he's throwing something, yes, *that* toy or item goes away. He hasn't hit anyone, so that's a plus.:) However, what I most focus on is trying to solve the problem at hand.
This means acknowledging "wow, you are really mad right now." and then sussing out the problem. Solving the problem FIRST then gives him the emotional space to be able to hear suggestions and corrections. "What could you do the next time your invention isn't working?" or "I know you don't like having to play in your room right now, and I'm cleaning the house and need you in here for a moment. How can we make this better for you?"
Here's something I want to address: your 2 year old does need more help with a lot of the self-care tasks, however, your five will need just as much help right now in handling their emotions. To me, when I witnessed my son doing this the first time during his fives, I thought "wow, it's like part of him is re-working at this challenging task of managing his feelings, kind of like when he was wee little." There are times when I'm pretty clear with him that "I need you to do this now, without throwing a baby fit" (i.e.-- getting ready for school, getting shoes on-- he really would prefer to stay home, but once he gets there, he's fine.)
I also have a 'blue dot' chart , where we put blue dots up to acknowledge when he's cooperative, has a positive attitude, or makes progress in something. "Wow, you got ready for school so cheerfully today with no fuss! Let's put three dots on your chart!" or "Thanks so much for making this easy for me--- I really appreciate your cooperation/help. That's four blue dots." We are liberal with them, and when he has 100 dots saved up, that's the time we go do something special with him, either as a family or one-on-one parent time. These are trips to the nickel arcade, a trip to a local inexpensive attraction, a special ride on the Zoo Train (we are members, so lower cost) , etc. The reward is time together instead of an object.
And yes, when he gets too, too much with his outbursts and we've tried problem-solving.... well, then he is walked to his room and "you may play in here until you've calmed down-- let us know when you are feeling better." Silly to say, but there was an episode of "Dinosaur Train" which had all the kids squabbling; the mother told them all to take a 'time out' and go find something to do which made them feel better. I find this kind of gentle guidance to allow THEM to find their own solutions instead of demanding that they change their attitude on the spot to please US has been far more effective at producing a calmer kid 20 minutes later on. The kid who has to conform emotionally, immediately, still hasn't had their unmet need met, so the problem is bound to come up again, and the child just feels incapable and perhaps even undermined.
Here's a link to that site I like, and they do have some helpful suggestion for this age as well. Good luck!
http://www.allthedaze.com/sdevelopment.html