The good news is that the Bluetooth freezing issue seems to have been ironed out after the Amber update on my Lumia 720. Originally, the phone used to freeze upon disconnecting my Bluetooth headset, but after the Amber update the phone seems to behave normally (almost). The main quirk that I noticed is the screen responding erratically at times (culprit seems to be the Facebook app).
I had initially thought that the Bluetooth freeze had been replaced with this touch issue, but some troubleshooting last night seems to suggest otherwise. The touch issue that I have been facing is that the phone does not respond to screen taps. Turning the display off and then on seems to fix the issue, but it can occur again at random. It can be especially annoying when you try to receive calls and the phone ignores your tap on the answer button. This issue does not seem to be restricted to the Lumia 720, as my mother who has a Lumia 520 also faces a similar issue.
Also at times, the phone seems to go on a tapping spree of its own resulting in false taps akin to having a keyboard key stuck. I have faced this issue when entering the lock screen pin resulting in invalid entries, and also having emails deleted due to the phone deciding to tap on the delete button on its own. It seems that the region just above the home\Windows button has this issue. Again, a turning the screen off and then on seemed to fix the issue.
I had initially suspected the extra sensitive touch capability of the Lumia 720 causing the problem (the Lumia 520 has this feature as well), but since I’ve never enabled this feature I ruled it out as a cause. The main culprit seems to be the Facebook app, as I had it controlling my lock screen backgrounds (same as my mother’s Lumia 520). I uninstalled the Facebook app yesterday, and gave the phone some time to behave itself. Things seemed to be ok, and I went ahead and installed the app once more. Things seem to be ok so far.
A couple of other quirks that I noticed with the Facebook app is that the built in Windows Phone chat for Facebook messages seems to have connectivity issues with the app installed. It happily showed me as online after I uninstalled the Facebook app. Things were back to the “Can’t connect” state soon after I reinstalled the Facebook app. The other quirk was concerning the album list in the Facebook app lock screen setting. I had set the custom album option and selected a few albums from my album list. However, the album list was missing the newer albums and this got updated after the app reinstall.
Bottom-line seems to be that the Facebook app has some issues that seem to get sorted out on an app reinstall (different issues on different platforms going by my Android experience). So, if you have the Facebook app controlling your lock screen backgrounds and are facing issues like me, try a reinstall.