If I buy a song from the iTunes store on my phone, will it automatically update to the library on my PC?

Or will it do it when I back up my phone to my PC?

Back when I had iTunes, it automatically downloaded.

Although that may have changed since 🙄

All iTunes purchases are tied to your iTunes/Apple ID. So as long as you're signed into that account on iTunes on your PC, it should automatically sync your purchases between your devices. I personally wouldn't count on it. I haven't used iTunes with an iDevice in a long time so I don't remember if it did that or if it still does. It can't hurt to try though.

Music doesn't download on its own. However, it will show up in your Library, but you will need to MANUALLY download it.