I've been using CarPlay since its inception. The Apple apps work very well (Maps, Music, Messages, etc.). However the third party music apps have been hit or miss for me since the beginning. I often use TuneIn Radio and I Heart Radio, but most of the time when I select them from the head unit, the app will open and I can't get anything to play. I can eventually get sound if I start playing the app from the phone itself, which defeats the purpose of CarPlay. I think Pandora is better, but the two apps mentioned above are pretty bad. 

One of my theories is that the ads that run on some music apps can cause playback issues in CarPlay. Does anybody have any insight?

I use Spotify and Pandora and on occasion have same issue as you.  Don't think its the adds though since I pay for Spotify.  Trying to play from the HU and nothing, unlock the phone and hit play/pause then try again and it works.  It's pretty buggy but when it's working it's great.  Haven't had any issues with phone/messages/calendars.  Maps works pretty good but not night mode for me at least.

Well, iOS 9.3 is supposed to add features to Maps and Music. Hopefully they can can bake some stability improvements in as well. 

I use Spotify and Pandora and on occasion have same issue as you. Don't think its the adds though since I pay for Spotify. Trying to play from the HU and nothing, unlock the phone and hit play/pause then try again and it works. It's pretty buggy but when it's working it's great. Haven't had any issues with phone/messages/calendars. Maps works pretty good but not night mode for me at least.

Apple needs to really beef up their 3rd party support for CarPlay. Only audio apps and they barely work probably because the SDK for it is shit. I don't even use any 3rd party apps for that very reason. First party apps on CarPlay are great though. Much better than anything else I've ever used anyways.

I like Apple CarPlay immensely. Most of the applications are well thought out and very useful although I still prefer Google maps to the maps application that Apple has incorporated into CarPlay but I really don't like to have to plug in everytime that I get into my car. I used to use my old 4th Generation Ipod Touch that was always connected to my AVIC-8000NEX head unit but now I have to plug in my Iphone 6S plus Whenever I get into my car which is annoying to me but that's the way it is. I wish that wireless CarPlay would become availiable. I would gladly upgrade to  a newer model Pioneer Head unit if it supported wireless CarPlay.