Home now... I have re-read all the posts on the other thread and your new one here. My understanding, and please correct me if any of this is not true, is that:
iPhone 5s had no issues on 8.4.1. On the other thread you stated "They reappear as 'open' upon powering on the phone".
What do you mean by reappear as open? Open on the multitasking window? The other fellow just had one App that would open automatically upon a restart, the Settings App. My understanding of his issue is that Settings somehow comes up by itself. Your case appears different. Obviously you can only see one App at a time on the screen so are you referring to 5 active in the multitasking window even though you closed them all prior to shutting the phone off?
You have tried a forced restart. That did not work. You then closed all Apps in multitasking and then did a forced restart again. Upon powering up the same 5 apps still appear in multitasking, is that correct?
If all this is true, then it is likely that your system (or an App) got corrupted during the update to iOS 9. Did you do the update over the air or through iTunes? OTA updates, while they work 99+% of the time, are not as reliable as updating through iTunes (OTA only downloads files that have changed, iTunes downloads all files). My suggestion at this time is to do a full restore as per the instructions below. A restore through iTunes will install a fresh copy of iOS 9.0.2 in your iPhone. Barring a hardware defect (unlikely since the problem started right after the update but NOT out of the question), a restore should put you back in near new condition. If it does not work, then your may have a defective device and would need to take it to an Apple Store or Authorized Dealer for evaluation. Hopefully the Restore will fix it.
Please follow the instructions in the article below. The process is rather straight-forward as iTunes will guide you along the way. You will have to disable Find my iPhone first. Don't forget to set it back to On when you are through (if it does not happen automatically) and also set Send Last Location to ON.
Connect your device to your computer. Then import all your photos from the device to the computer and copy all important documents directly to your computer as well. Then do a full backup to iTunes, just follow the instructions in the article. Then select Restore iPhone and iTunes will install a fresh copy of the current iOS and will erase all your data and settings. After iTunes has installed a fresh copy of the iOS, it will ask you if you want to set your device as New or restore from Backup. Choose From Backup. Restoring from backup will bring back all your Settings, Apps, Photos on Camera Roll, etc. You should not lose anything.
After completion check to see if works ok now. There is a possibility that restoring from the backup may bring back the system/app corruption. If this occurs (unlikely) you would have to start over again, set it up as New and install all Apps manually (a time consuming process). You would also lose your photos and documents so it is important to back them up first as I mentioned above. I have done this procedure several times in the past and restored from backup with no difficulties - it takes 30-50 minutes. One time however, it had to set my iPad as New due to a Photos App corruption issue - it took me quite a while to then reconfigure all settings and apps again so be warned that this, while not likely, is possible. Good luck.
Use iTunes to restore your iPhone, iPad, or iPod to factory settings - Apple Support