Google Play Music question/synching music

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!

 

What do you mean save them for offline use? Is that wireless or through usb? The former is what I'm wondering if it does, as it seemingly did it for the first two songs I added to the library besides the ones I already had on my device.

 

That's exactly what I needed to know. You my friend are the best. Thank you very much.