The Arduino Micro requires 5V. It runs at 16MHz, and you can't run at 16MHz at voltages below about 4V.

To run off a 3.7V battery you need an Arduino board that runs at 8Mhz, such as the "3.3V Pro Micro" from Sparkfun.

To know if your battery is suitable for your project you need to work out the hourly average current draw. That is the proportion of "active" and "inactive" times for each of the devices (transmission duty cycle for your transmitter, sleep times for your MCU, etc). Until you know that you can't know the capacity (mAh) you need for your project to run for the desired amount of time between charges.

The Arduino Micro requires 5V. It runs at 16MHz, and you can't run at 16MHz at voltages below about 4V.

To run off a 3.7V battery you need an Arduino board that runs at 8Mhz, such as the "3.3V Pro Micro" from Sparkfun.

To know if your battery is suitable for your project you need to work out the hourly average current draw. That is the proportion of "active" and "inactive" times for each of the devices (transmission duty cycle for your transmitter, sleep times for your MCU, etc). Until you know that you can't know the capacity (mAh) you need for your project to run for the desired amount of time between charges.

You can also drop the idea of an Arduino and branch out. A board that is probably tailor made for your project is the Adafruit Feather M0 Bluefruit LE which is designed to run from a Lithium Polymer batter (and even includes the charger) and has bluetooth built in.