1. ESP8266 is reported to draw up to 215mA during transmissions while the 3.3V voltage regulator on Teensy can provide only up to 100mA. It should work but not very reliably. It is recommended to use an external battery to power the ESP8266.

2. You should start by getting ESP8266 to accept incoming TCP connections (running a TCP server) and check if you can connect to it from your local network. Afterwards you need to configure your internet router to forward the incoming connections to the appropriate port on ESP8266.