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You could try a couple more things as you have mentioned you have tried some of the obvious......running it in Safe Mode for a while & check if it runs smoothly or due to the fact you have just downloaded the new firmware i would try a repair thru XC
http://talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008
http://support.sonymobile.com/global-en/xperiaz3/tools/xperia-companion/
There's no recovery option
Frankly, Nothing works. We should wait suffering, till we receive the next update. Even the so called heating issue which they have claimed, to be fixed with this update, is of no use. My device still gets heated. And even the saved playlists in Music album are also vanishing in my device. These people at Sony are horrible in software front. Infact they have revamped the entire user interface in Marshmallow which was a welcoming thing, because its light and fast. But its no the complete version like the lollipop. Many features and options of lollipop are missing in this new revamped UI.
Do you by chance have an SD card inserted?
I had very similar problems a while ago, and it was because I used an SD card that was much too slow for the phone (SanDisk Ultra). It worked fine for a while after the card was formatted, but would start bogging everything down again after 2-3 days of use. That could have just happened to coincide with your firmware upgrade.
It affected mostly apps that used the SD card (loading streams in the album and camera apps, for example), but it also noticeably slowed down other apps and the OS itself. Sometimes it took 20 seconds to just unlock the phone. Everything worked again as soon as I removed the SD card (and now replaced it with a faster one).
I can't reproduce this issue with my Xperia Z5 Premium, repair the phone with XC and report back
Alternatives on How to backup Xperias
from this video link we can see that the flashlight icon cannot be displayed, the only solution is restart the Z5 (32.2.A.0.305 Singapore).