ESP8266 Community Forum

I've been trying to set up an ESP8266-01 with an Arduino Nano, using the instructions in the link below.

https://elementztechblog.wordpress.com/2015/05/13/esp8266-based-temperature-data-logger-using-arduino/

Somehow, as soon as I connect the ESP8266 to the Arduino, the COM port connecting the arduino to the PC is not detected, and I cannot compile it. Is there some library or driver I am supposed to install beforehand?

Ask your question over there; this is a forum about the ESP itself, not about the Arduino.
Most sites/people that promote using a combination of an Arduino with an ESP like that (using AT commands), don't really now what they're doing or their info/knowledge is outdated.
In 95% of the projects there's no need for an Arduino, since an ESP alone would suffice.

There's nothing wrong with Arduino, in fact it's a great board, but using AT commands is cumbersome, pointless and it makes things unnecessary complicated and hard to debug.
An ESP can be programmed using the same tools and code as an Arduino (using the Arduino IDE) and it is 10 to 20 times faster (160MHz vs 8 or 16MHz for the Arduino).

There are excellent projects that exactly do what you want and in a proper way (although I indeed see a lot of the same "Wrong" Arduino/ESP combo's out there...).

Another tip: since you're new, get yourself an actual development board (they're only about $2.50, see my signature for examples) for a flying start.

Somehow, as soon as I connect the ESP8266 to the Arduino, the COM port connecting the arduino to the PC is not detected,

The 3.3V from the Arduino is not capable of powering the ESP. Also the example you are using is bot doing any signal level translation. The Arduino port high voltage is 5 volts but the ESP should only take 3.3 volt levels. Just because it is well presented doesn't mean the information is correct.

Get yourself a proper board to work with. It will save you a lot of frustration.

... and that project uses a LM35 for temperature sensing and therefore needs an ADC input. Not a great choice nowadays.

Many sensors can be obtained with a digital interface. A DS18B20 temperature sensor, for example, is accurate and needs just 1 digital input to support it. LIbraries make it very easy to use. It is also very cheap, or a little bit more if you want it in a probe housing.

See, for example, https://www.instructables.com/id/WiFi-T ... h-ESP8266/ which actually discusses both LM35 and DS18B20, Note that I don't think Step 8 in that article is necessary for most boards.