Apps not on iTunes library but on iPhone 5S

Note: when I first created this message a screwup on the keyboard posted the message before I had written more than two sentences.  I hope no one responded to my incomplete message.
For some time now when I elect to install a new iOS release (not at 7.1.1), iTunes always tells me (both iPhone 5S snd iPad Mini) that there are apps on the iOS device that are not from my iTunes library?  Since I install all apps throught iTunes or from the iOS device purchase, how can this be.  I have never broken into iOS to install except by the standard accepted Apple process. 
So far, each time, I cancel the update, do a full backup of the device to my desktop computer iTunes (version 11.1.5 (S)).  This time after the backup sync, iTunes transfers 56 purchased apps from my phone according progress status in iTunes. 
Several questions:
   1.  How can I get a list of all the apps in my library AND all the apps on my iOS device, so I can know what I've got in case things get corrupted so I can do a complete restore.
   2.  Why is it reporting apps that are not in my iTunes library.  Every app on my devices were purchased in ITunes on my iMac desktop (OS X 10.9.2) or from the app store icon on my devices home screen?
Without the list I asked for, I'm afraid taht if things go 'south' I won't be able to get back all the apps I purchased.  There are so many apps in the store (not a compllaint but an observation on the size of the task), that I have no idea how I would know which are missing.
How can I get this fixed so I can backup my devices and be confident that I won't lose things.  I allow backup from WiFi (my personal WiFi in my home) when the iOS device is attached to the power.  This happens nightly for my iPhone when I put it in the charge by my bed.  And at  least weekly for my iPad which I plug in as it needs charging. 
Thanks,
Bruce

Well, your post is very long, so I didn't read the whole thing in detail.
But here are the answers to your two questions.
1.  You can see the apps in your iTunes library by clicking the "Apps" tab toward the upper left of iTunes (it may say "Music" to begin ... then there is a drop down list).  You can see all the apps on your iOS device in one of two ways:
a.  Swipe through the various home screen pages on your iOS device and note them.
b.  While your iPhone is plugged into iTunes, click the "Apps" tab behind the iPhone's summary tab.
2.  Assuming every one of your apps was purchased using the same Apple ID and password, apps that were purchased or updated on your iOS device will not appear in your iTunes library until you back up the iOS device.  You can get apps that you purchase on your iOS device to download automatically to iTunes by doing iTunes > Preferences > Store > Automatic Downloads > Apps = "check" from iTunes.  But I don't think this would capture app updates.

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