Firefox on El Capitan cannot create profile in standard folder
This issue, as well as issues I was having with iTunes and f.lux, were resolved by resetting permissions on the home folder. Before doing this, I had upgraded to Sierra from El Capitan. So the reset process I used is for Sierra. It's included below, and the original can be found here:


Resolve issues caused by changing the permissions of items in your home folder



  Reset permissions
  
  If the issue occurred after changing the permissions of items in your home folder, follow these steps to reset the permissions:
  
  
  From the Finder menu bar, choose Go > Home. Your home folder opens.
  Choose File > Get Info. An Info window for your home folder opens.
  If the “Sharing &amp; Permissions” section at the bottom of the window isn't open, click the triangle in that section to open it.
  If the Lock button at the bottom of the window shows a closed lock , click the lock and enter an administrator name and password.
  Click the Action menu  in the bottom corner of the window, then choose “Apply to enclosed items.” Click OK to confirm the action. A progress bar appears at the top of the window.
  When the progress bar completes, open the Terminal app from the Utilities folder of your Applications folder.
  Paste or type this command in Terminal, then press Return:

diskutil resetUserPermissions / `id -u`

  
  On U.S. keyboards, the ` character is just above the Tab key. After entering the diskutil command, if Terminal says that permissions reset on user home directory failed (error -69841), enter chflags -R nouchg ~, then enter the diskutil command again.
  When the process completes, quit Terminal.
  


I discovered a quite different process for El Capitan on The Mac Observer OS X: Repair Home Folder Permissions, which I have not tried, and so cannot speak to its efficacy.