I've got a 520 and now the 520+ and they work like your friend's device -- when I save a ride on the Garmin unit, it automatically uploads it to Strava if my phone is nearby.

Someone else may have better troubleshooting tips, but this is a sequence I'd try:

1. unpair everything, have your phone forget the Garmin on bluetooth. Turn off your Garmin. Uninstall the Garmin Connect app. Leave everything off for now.
2. Update your phone's iOS, reinstall and update Garmin Connect.
3. Turn on the 520, pair it with your phone and let it update its software
4. Pair Garmin with your Strava
5. Try the 520 just with your phone, see if things are working
6. add the heart rate monitor, see if things work
7. Add the speed/cadence stuff, see if problems develop.

Lastly, the 2 bluetooth connections serve two different purposes. One is for all the cycling functions. The other is to send your texts and call alerts to your 520. I never have the second one, for calls and texts, on.

Hope that helps. When all else fails, there's always the manuals.