Does iphone have an activity/ diagnostic log
Yes, but it's just a crash log...

Settings > Privacy > Diagnostics &amp; Usage > Diagnostic &amp; Usage Data

However, my experience is that it's more likely that your headphones have a fault in the cable or inline remote or dirt in your headphone jack or dock connector (pre-iPhone 5) which is causing it to pause randomly.

This is very common, especially if the headphones don't sit fully into the jack because of lint at the bottom.