What to buy for a project with an ESP8266 WiFi module? Versions, power supply, programming

If you are not going to use the other outputs from the ESP, then the cheapest version will suffice. If you want to play with some sensors, use them for more advanced projects, then the second option. The price is similar, so I would take the second option personally.

Do you mean to power the module already after uploading a properly prepared program/script? If so, then you can buy one with the battery cage soldered on, or you can buy the cage separately and rework it accordingly so that you connect it like a power supply under the module, but that will probably come out more expensive. If that's not what you're talking about, you can buy a power module that plugs into a contact plate.

You program the ESP over USB using a UART-USB converter. You can program using the NodeMCU, Arduino IDE, or MicryPython.

It's mandatory that you have cables to connect all this, unless you already have them and didn't write about them , because it's obvious, passive components. I guess that's all of such basic stuff.