Can't get ESP8266 working
Personally I think using an ESP as a interface for an Arduino is backwards and the Arduino ought to act as the interface to your devices for the much more powerful ESP.

I think you had a board that was configured to work as a interface between serial and Wifi, not as a normal board.  I'm pretty sure that you have now shagged it totally by trying to write programs to it.  However you might be able to recover it if you can find the original program and upload that to it.  You don't need the ability to program and ESP-8266 in the IDE to use this board, you can not upload sketches to it, you can't access the GPIOs.

I think you should just be able to slide the switch to "program" power the board up and type AT commands via a terminal emulator to set the board up and then power down, slide the switch back to UART and away it goes.  As to the Tx and Rx pins, that depends on weather its saying "this is the Rx pin" or "this is the Rx connection", you'll get it right second time :)

If you want a ESP8266 try AliExpress they cost between £1 and £2 and I've never had any problems, probably because they were made in China.