Hi,

The way I transmit data with the ESP8266 is by connecting ESP8266 and the arduino via the serial port, and the ESP8266 runs a simple web server.
See both source code:
https://github.com/babelouest/taulas/tree/master/taulas_esp8266_webserver
https://github.com/babelouest/taulas/tree/master/taulas_tls0

If you want the ESP8266 to transmit data, not listening to incoming http request, you can use the library ESP8266HttpClient, which I don't know that much, but some examples are available:
https://github.com/esp8266/Arduino/tree/master/libraries/ESP8266HTTPClient

Hello,

The number of sensor connected is limited by the number of IO pins you have available on your arduino. There may also be CPU limitations on the arduino to process a large number of sensor but I never reached it out myself. You can try by adding sensors one at a time and check if it's still working.

If you want to transmit data via the ESP8266, that's possible using the client API, I didn't do it myself but I've tested it and it's very simple to use, check the documentation:
https://github.com/esp8266/Arduino/blob/master/doc/esp8266wifi/client-examples.md
https://github.com/esp8266/Arduino/blob/master/doc/esp8266wifi/client-secure-examples.md

I'm not good at electronics schematics but I found a picture of the connection type I use between the arduino and the ESP8266:
https://raw.githubusercontent.com/mrizvic/nodemcu-uart2mqtt/master/WiringDiagramEsp8266.png

They communicate via the serial bus.

To check the ESP8266 connections, I add 2 leds connected to the pins available on the ESP8266. The first one lights on when the Wifi is connected, the second one lights on when the communication with the arduino is established.

I will try to find some time to make a full schematics, but there are documentations available on the internet to help you program all parts.