Thanks for the detailed reply.
First, I have a Mac Mini also, it is a late 2012, 2.3 GHz Intel Core i7 model (I also have a mid 2013 13" Mac Book Air).
Secondly, glad you are making backups to an external device. Myself I use SuperDuper! to back up each of my machines to 2 separate external drives. The main advantage of such a backup is that it is bootable, as it makes an exact backup/clone of your entire system.
Speaking of that, and Disk Warrior, I also have Disk Warrior (and Tech Tool Pro too), but I have Disk Warrior as a disk image. In order to have it repair my internal drive, it must be launched externally, and I can do that via the SuperDuper! backup/clone. (TechTool Pro is somewhat more "ingenious", in that it creates an eDrive on my internal drive, and I can launch Tech Tool Pro from there, and that acts like it is being launched externally). So, how did you run Disk Warrior?
As for repairing permissions, Disk Warrior (not Disk Utility, nor Onyx (I'll describe that software below)) does not automatically repair permissions (Tech Tool Pro has that feature also, but again, it is not automatic). The newest Mac OS, El Capitan, "claims" to repair permissions "automatically", but I use (and have relied on) the excellent freeware program called Onyx to do that. And, that program has other useful features.
Regarding disk cleanup, one needs to do that "on their own". That is, regularly clean off/remove unneeded files/folders, and use some software to perform similar functions also on a regular basis. Myself, besides consistently removing unneeded files/folders, I use Onyx to clean out some stuff (log files, cache files, etc.), and Tech Tool Pro. Disk Warrior can clean stuff also. Again, though, it is best to do that on a regular basis. My use of Onyx and TechTool Pro (and SuperDuper!) is once a week, on a Saturday (I also first go through a procedure to permanently remove deleted EMails in Outlook 2011). In fact, I will begin that process soon, for both of my machines.
With the possible incomplete download, you might have some directory damage, and Disk Warrior (also Tech Tool Pro and Disk Utility) can repair that. If you can, use Disk Warrior to do that first. Then, given that you have a backup, you could try to apply the OS 10.9.4 Combo Updater. If that does not work, then you should consider doing a clean installation of OS 10.9.4, and then use Migration Assistant to "migrate"/copy all the "stuff" from your Time Machine backup. (Make sure to save the OS 10.9.4 Combo Updater on your external drive (unless you first do another Time Machine backup). To do that clean installation, you will need to boot to the Recovery Partition and launch Disk Utility from there to 1) Erase, Format, and Partition your internal drive, and 2) do a clean installation of OS 10.9. You can then boot your machine via OS 10.9, and apply the OS 10.9.4 Combo Updater. Then, use Migration Assistant to get the rest of your stuff. This link has information on that (invisible) Recovery partition:
https://support.apple.com/en-us/HT201314
Good luck!