Question about Google Play Music and synching songs

Just had a few questions about Google Play Music, as I just started using it and I'm not 100% familiar with it yet.

First off, do the songs added to the cloud (through my computer) sync to my phone? I know they can be played off the cloud, but do they actually download to the phone if connected to wi-fi or something like that? I didn't think so, but then two of the songs I added recently to the cloud are now on my device (not through the cloud, actually on the device) but now I can't seem to get any other songs to transfer over. I have no idea how the original two got on, and I definitely don't remember doing it manually so I just wanted to know if someone could help me out with this.

Also, do songs I put on my phone, transfer to the cloud automatically? Basically, my first question but backwards. This I don't care about too much since I rarely get music on my phone before my computer but I was just curious.

Hoping someone can answer this for me,
Thank you in advance!