The BB processing is part of the receiver so yes, it needs to be on in order to receive anything. What the RF does is select, filter and convert down (in frequency) the BB signal. You still need to demodulate the signal which is what the BB processor does.
There are several ways to implement a BB processor. That can be in software, in hardware or a combination of the two. If (part of) the BB is software based it can have its own core but that is not required. It depends on the design implementation.
Most BB processors do use RAM and mostly this will be embedded. But it depends on the application. Simple protocols (for example Bluetooth 1.0) require much less processing power than a complex OFDMA signal (for example LTE, Wifi AC).