T.,
This sounds soooo familiar! I also have boy/girl twins. However, like Teresa, I had no issues with letting them climb and jump off things. I have pictures of them on the dining room table at that age! My philosophy is that it is a normal kid thing, they're going to do it anyway and they're building their gross motor skills, as well as judgment about their own limitations. So what I did was make some places "okay" to climb on and jump, and taught them the proper way to do it (i.e. if it really was too high to jump from, I taught them how to go turn around and go down feet first).
But when they tried to get up somewhere that wasn't "okay" (like the kitchen table or counters), I'd tell them it wasn't allowed, and redirect them to one of the "okay" places. They quickly learned what was allowed and what wasn't because they still had an outlet for their need for the activity in some form...and they couldn't push Mommy's buttons on the issue.
It was easier for our son, who was born with an excellent sense of what he could and could not do with his body. Our daughter had no sense of fear whatsoever and had to learn caution from a few hard knocks. When she started going for higher jumps and knew that she could get bruised, she'd ask for me to hold out my hand for support so she could use me for balance while she taught her muscles what to do.
My daughter was the same with tantrums (still is sometimes at 3 1/2)...long, loud screeches that never seemed to end. Instead of completing ignoring it at that age, I'd reflect back what I thought she was thinking/feeling "I know, I know. You're so mad and angry. You wanted to play with xxx and Mommy took it away. You don't understand why you can't play with it. Mommy's so mean sometimes. It's so frustrating not to be able to do whatever you want when you want." etc. If she wasn't listening to the words, then I'd hold her and repeat them over and over until she calmed down. I think you might not be giving her enough credit; she probably does understand what you're saying (and at those moments she doesn't care for what she's hearing) but she might be feeling you're not understanding her anger/frustration.
As she got older and better able to express what was upsetting her, the tantrums decreased in intensity and duration. I tell her that she has to use her words to tell me what's wrong because I can't understand them when she's whining.
Oh, we also went through a stage with our son and tantrums at bedtime, but he got physical and tried to bang his head on the rails of his crib. When he did that, I'd stop him, tell him he wasn't allowed to deliberately hurt himself and warn him that all his books and friends would go away if he kept doing it (he used to sleep with a dozen books and a half dozen stuffed toys). Then I followed through and only allowed him to keep his absolute favorite lovie and emptied out the crib. He got items back the next time he went to bed without a tantrum. After two cycles of this, all I had to do what ask him if he wanted his books and friends to go away again and he'd still cry, but tried to self-soothe and wouldn't try hurting himself.
Now when he and his sister sometimes feel out of control, they'll get on their hands and knees and slap their hands on the floor 4 times (I have no idea why the number 4) and then get up to tell me that they "bumped" their hands and want the Mommy boo-boo kisses. And the mini-tantrums cease as soon as I "comfort" them with kisses for the boo-boos. I have absolutely no idea why it works for them, but I think it's an acceptable compromise!
I hate to tell you this, but I found 2 to be a much easier age than 3; at 2 you can still redirect them fairly easily. At 3, they remember what they were doing wrong and keep going back to try to continue! LOL!
Truth is my husband and I have never successfully been able to use timeouts at home, (maybe because it doesn't occur to us to try very often!) only out in public where we sit them down and face them at their level and tell them they have to stay put.
Don't be surprised, though, if what works for your son, won't work on your daughter and vice versa. Good luck; you'll get through it all and find what works best for all of you.
Note: Just read Marie's response and that works for us as well, letting them know in advance about a transition (i.e when Mommy takes a shower, we'll all get dressed and go shopping). And reminding them along the way. "Okay, Mommy's had her shower, it's time to get dressed so we can go shopping." The 'bye-bye' trick works too; I think it gives them a feeling of control.