If you refer to CIP-8 chip, as the data sheet indicates, it work from 3 to 5.5 volts. So, yes, 3.7 is fine.

Now regarding the transmitter, you can find modules that work from 1.8 up to 5.5 volts. Bruce does have such modules (don't know if they are from 1.8 or 3). I have found from Dorji.com modules that are working from 1.8 volts and up to 17dbm power @5V.

What you are asking is easy to do, and since you don't want to identify each transmitter, every one transmitter will have the same ID.

Your switch will be controling the power to the whole transmitter circuit, so when the switch is open, you current drain will be zero.

On the receiver side, the current will be in the range of 10-15mA, so I suppose your battery will not last for very long. You want a line power supply, or much bigger bat.

What is your location, USA?

Ioannis