Q: Why does Itunes sync unchecked songs to my iphone 6s, while also refusing to download some of my checked songs?

Hello gfazzy44,

I understand your frustration with syncing your music to your iPhone. Let's see if we can figure out what is going on. 

Now you posted this in Apple Music section. Are you using Apple Music? If you are, then you don't need to sync and all you need to do is enable iCloud Music Library. This will allow your iPhone to see your entire music library that is on your computer. Then it would be a matter of downloading the tracks that you want to store on your iPhone while others would just stream. Take a look at the article below for more details about this. 

Sync Apple Music across your iPhone, iPad, iPod touch and computer
https://support.apple.com/en-us/HT204926

If that is not the case, let me know.

Take care