Yes, it would. The Wixel draws a few tens of milliamps while transmitting and it can go down to just a few microamps while it is sleeping. You can see the CC2511F32 datasheet to get more specific numbers and also:
Yes, there could be a delay of several hundred microseconds. The biggest factor is when you choose to calibrate the radio’s frequency synthesizer. It might be possible to calibrate the frequency synthesizer before going to sleep so you don’t have to do that step when you wake up, but I’m not sure. More information about the radio’s state machine and the delays between different states is in the CC2511F32 datasheet.
–David