Can a nRF24L01+ chip work alone? (i.e. without Arduino borad)
The nRF24L01+ is indeed just a tranceiver IC with no embedded firmware you can modify.

It has some capability to reduce the amount of work needed by the application controller (which can be any MCU, you can build your own ultra low power design if an Arduino is too power hungry), but the protocol stack has to be implemented there.

Maybe you want to take a look at the nRF24LE1 - which has integrated flash (16kB) and MCU (8-bit), but I don't think it's possible to fit a wifi stack in there.

There are similar solutions from TI and others around, which might be worth a look. Wireless stuff is advancing very fast these days, so I won't post any more specific stuff, as it will be obsolete on Monday.