Unable to update or download apps on iPhone 6 - process hangs
It seems you're in a tight spot and have a pretty tight knot tied up with some really solid thinking.

Let's unravel things a bit. Get out a piece of paper and start writing times and limiting things you change to one at a time.


Put the phone in airplane mode and power it off. (note what you did and the time)
Power it on while disconnected and go delete all the apps that are stuck that you can afford to lose and re-download. (keep writing times and actions :-)
Check your storage usage and make sure each app gives you a number. If you can't get the usage numbers to paint while in AirPlane mode, start deleting things that take a lot of space. Movies, songs you can re-download until the device is working.
If you can't get it to a sane place in AirPlane mode, consider a call to AppleCare or just getting a backup to iTunes and wiping the device.


If you can resolve the transitory apps and get things working, go ahead and re-connect to the network and see if you can let the rest of the apps download. If so, proceed:


Turn off all background refresh in settings > general
Turn off Handoff (same place - higher up)
Set the time wrong - about 4 minutes in the past
Enable automatic time setting and verify that it returns to the correct time
Reset network settings (which will clear up several things and perform a minimal restart)
Pick one app that's small. Clicker is a good one Try downloading it.
If that fails, sign out of the App Store and power off the phone
Start the phone and try to download Clicker again.


If that fails, Apple support might be needed to assist. You could try changing your network (use LTE or cellular data if you're on WiFi now) or change WiFi to another location. The good thing about your notes is you will have some very specific times and steps and can start to notice patterns and perhaps ask a follow on question if needed.