You would need to set the time from NTP as the RTC is just a counter, and memory that survives reset.
Most of your questions can be answered by looking at the document here.
I think the ESP can wake from sleep using a GPIO pin when in light sleep.
Why the need for sleep mode? Are you running from battery and need them to last as long as possible or some other reason. Not knowing your setup, if it's battery power then maybe a different system/MCU would be better suited.