I Have an iPhone 5S that started to have both problems as well. I find it interesting that so many folks have posted with so many different types of fixes. I tried one of the fixes, but then my apps stopped updating. Then I tried a hard boot, which also didn't work. So I decided to reset the network before going to the extreme of downloading my entire iPhone's contents onto my laptop, which is a significant amount of information. I have iOS 8.4.1 currently, though the problem could have started with an earlier update and I didn't notice until I tried to install a new app. My iPhone capacity is 55.9 GB (not counting iCloud)
and I currently have 6.3GB left. I have used 49.6GB. The salesman at Verizon assured me that I couldn't possibly ever fill my new phone back when I purchased it. I have 11,980 photos, 1,890 songs and 90 apps. I have yet to figure out how to go about removing a number amount of photos to free up some space (a question for another place in the forum). To reset the network I went to Settings>General>Reset (at the bottom of the screen if your iPhone is like mine)>Reset Network Settings (in the middle of the screen). For me this worked and it helped me to avoid having to do a backup of my iPhone (crazy to think about!). Maybe this will help someone. This problem seems to have such a variety of fixes that it's impossible to predict what will work. I'm in agreement that Apple did something in one of the updates to cause the problem because my iPhone worked fine up until the resent updates. Good luck to all who end up in this forum looking for help!