OP- I love your username, I just moved back from a few years in Japan and it reminds me of being there!
I think that 4 is a really hard age. Most people I've met will agree. When you have a kid that age and an infant it's compounded. There is a tendancy to look at your 4 big year old and look at the tiny helpless infant in your lap and think- "this kid is NOT a baby anymore, she shouldn't act like one, she's a kid..." The problem is that while she certainly isn't a baby, she's not a real "kid" either. Preschool age is a struggle,the change-over from toddler to kid is a long process, it sputters along. It's only in retrospect, when I can see how different my now 6 year old is from my then 4 year old, that I can appreciate that difference. And I wish that I had that clarity then, because I was too hard on her and I expected too much at times, which made it hard on both of us. My 6 year old does, pretty much, what is expected of her. She will clean her room or put away her clothes, or help her sister. At times I have to nag, but she does it. The same girl, at four, would do things like that if she was in a mood to help, or if I could make a game of it, etc. But if I tried to mandate it, it would turn into a screaming fest or a tantrum. It wasn't worth it.
I think you'd both be much happier if you let the small stuff slide when you can. The pants, the window, etc. Some tantrums you might not be able to avoid. If she can easily get the spoons everytime, has no problem reaching and opening the drawer, then I would have said, "you want a spoon right now, you can get it, or you can wait until I can do it later." And maybe a tantrum would ensue, but that one tantrum is an an improvement over 4 or 5, right? If you can focus on making her comfortable in her new place in the family, it will certainly help some of the problems your having with her behavior. If you can temporarily deal with a bit of mess and household that is a little more chaotic, while you basically hang out with kids, I think you'll find that she'll soon be more willing to help out and act like a helpful "big sister". Good luck!