From my experience with one babe (but a feisty little guy!), I would agree that the 4 - 7 month period is a difficult one. Ezra's behavior during that time was almost precisely like your DD's, and I also found the "breast grabbing" to be the most difficult. I always envisioned breastfeeding to be a relaxing and comforting experience (for both mom and babe), but even with my best attempts to hold his hands, I would often end up with bruises.
Sometimes giving him fabric to hold or singing lullabies to distract would help. Generally speaking, he was most animated while around me and DH. So, taking him out for walks or trips around town would often calm him for a time. While at home, I would keep him in a constant rotation -- from sling, to crib, to exersaucer, to pack and play, etc -- about every 20 minutes or so.
Ezra just turned 8 months a few days ago, and he has begun moving everywhere. Just today, I realized how much calmer he has become, even when we are holding him. He still screeches very loudly at times (out of happiness or sometimes annoyance), but he doesn't seem nearly as "grabby" or hyper. So, perhaps you can find some comfort in the "phase" idea. We've moved on to the "restless sleep" phase, so I'm trying to have hope myself.
If you can't connect your DD's behavior to any food (a definite consideration), perhaps all of her pent-up energy will be released once she can get her little limbs moving around!