Which chipset does your broadcom have? BT3b works out of the box and supports injection for at least all bcm43xx chipsets.
PWR readings are not supported for broadcom chipsets so this is nothing to be worried about.the problem is when i airodump-ng (interface) all the network i can i see have PWR 0 and some even -1 which is quite strange because generally i get good signals from my neighbours and excellent signals from my own network.
Naturally you can not expect to stay connected to a network while putting your wireless card into promiscuous mode. And as a side note, do not use the Wireless Assistant but learn to use the command line instead and you will be spared from a lot of connection issues.Also I noticed that while i do some airmon-ng commands i get disconnected from the internet and cannot see ANY wireless networks in the wireless assistant all of a sudden.