Lots of good questions! Some vegetables can become more nutrient dense when cooked or frozen compared to raw veggies. However, to answer your questions:
Any time a vegetable is stored whether cooked, raw, or frozen it may be prone to some vitamin nutrient loss, but not minerals or fiber. However, this would be slight and probably not that significant from a nutrition point of view. For example, if the vegetable provides 130% of vitamin A, and it decreases to 127%, when you only need 100% anyway, does it really matter? In fact, as a matter of convenience, I personally like to blanch the entire bunch of broccoli raab so that it is ready to sauté or stir-fry for dinner during the week. Making sure that it is easy to use and gets EATEN, to me, is far more important than any minor nutrient loss that might or might not occur from having been cooked and then stored in the fridge.
It is entirely up to you if you want to freeze what you are not eating, but if you are going to freeze the vegetables, try blanching them first.
Leaving cooked veggies in the fridge for 1-5 days should be fine (I do it all the time).