Can packets be read out of the NRF24L01+ while in RX mode?
Yeah, that is indeed confusing. It has been a while since I read that document (I am more familiar with the RFM73 clone), but note that sentence states configuration registers, while 8.3.1 talks about command and status registers. 

My best guess is that the limitation applies to a small subset of the registers, probably only the CONFIG register. Or it is a sentence that was left in the datasheet totally by accident. Note that it contradicts the 3d paragraph of the introduction.

I have used this chip briefly (mainly to test whether it is compatible with the RFM73 - it is), but I never had to put the chip in standby to do reading or writing of the packets.