If you want to send real audio from the phone to the esp32 using udp, you will have a lot of work to do.

If you want to send a udp datagram (text msg) to the esp32 you can use a Text-To-Spech (TTS) library to play it to a speaker

What esp32 cam do you have ? Most of them dont have the free pins required for the display ?

What esp32 -s2 module do you have ? and what i2s library, does it support esp32-s2 ?

No, OTA is using WiFi and then you can not use deep sleep. In you code you can setup a funktion to run when OTA starts, set a variable to true and then disable everything else when it is true.

With only one 18650 battery, be sure to use high quality , some cheap from china does not have the required power.

I have esp32 running on one cell, so with the right shield and battery it will work as Long as the battery have Power.

Depend on how long you need, on Banggood they have 2 lenght, one short (normal size) and one longer.

You can find example code for UDP server for both esp-idf and arduino, you can use in eclipse idé, use Google to find it.

UDP might not be ok for handling your audio, if that the case you can use TCP.

Use a small esp32 in Deep sleep, with a battery and button The esp32 wakes up, when the button is pushed and send an email, or TCP/UDP message to other esp32 ( Places within WiFi Range) which the flash red light, activate a buzzer, promt message on a max7219 display et.

Have you selected the right module in Arduino and running the latest version ?
If yes I have No clue .