After Upgrading to 10.9.4, Apps Don't Quit, Won't Restart. Now What?

Question

I have a mid-2011 Mac Mini. It was a joy to use until OS X Mavericks came along.
I'm on the road a lot, and remote into my Mac on a daily basis, whether it be from my iPad or laptop.
Since upgrading to Mavericks, I've had to reboot the Mac about once a month. Since 10.9.4, it is about once a week.
Last night, I closed Mail and Chrome, but they would not exit. I had to use "Force Quit".
Today, neither will open. after fiddling with PS AX from the command line and trying to kill the tasks, it seems that "Force Quit" didn't finish the job. The task status now appears as "?E" or "?Es" when using PS AX, and the KILL command doesn't do anything. The process names are now in parenthesis.
I tried "KILLALL Finder" to see if flushing-out the cache would help. Now, Finder won't start.
Before this started, I took a peek at the Preferences for GoToMyPC. Now, System Preferences won't start.
Most notable is that the Speech Recognition Server crashes a couple of times a minute.  Since updating to 10.9.4, the round speech control no longer appears on the desktop. I didn't notice it at first, because when I'm away, I can't hear my Mac talk to me, anyway.
So, tomorrow, I've got to make a six-hour road trip to boot my now buggy Mac.
<Edited by Host>

Answer 1

Please read this whole message before doing anything.
This procedure is a test, not a solution. Don’t be disappointed when you find that nothing has changed after you complete it.
Step 1
The purpose of this step is to determine whether the problem is localized to your user account.
Enable guest logins* and log in as Guest. Don't use the Safari-only “Guest User” login created by “Find My Mac.”
While logged in as Guest, you won’t have access to any of your documents or settings. Applications will behave as if you were running them for the first time. Don’t be alarmed by this behavior; it’s normal. If you need any passwords or other personal data in order to complete the test, memorize, print, or write them down before you begin.
Test while logged in as Guest. Same problem?
After testing, log out of the guest account and, in your own account, disable it if you wish. Any files you created in the guest account will be deleted automatically when you log out of it.
*Note: If you’ve activated “Find My Mac” or FileVault, then you can’t enable the Guest account. The “Guest User” login created by “Find My Mac” is not the same. Create a new account in which to test, and delete it, including its home folder, after testing.
Step 2
The purpose of this step is to determine whether the problem is caused by third-party system modifications that load automatically at startup or login, by a peripheral device, by a font conflict, or by corruption of the file system or of certain system caches.
Please take this step regardless of the results of Step 1.
Disconnect all wired peripherals except those needed for the test, and remove all aftermarket expansion cards, if applicable. Start up in safe mode and log in to the account with the problem. You must hold down the shift key twice: once when you turn on the computer, and again when you log in.
Note: If FileVault is enabled, or if a firmware password is set, or if the startup volume is a Fusion Drive or a software RAID, you can’t do this. Ask for further instructions.
Safe mode is much slower to start up and run than normal, with limited graphics performance, and some things won’t work at all, including sound output and Wi-Fi on certain models. The next normal startup may also be somewhat slow.
The login screen appears even if you usually log in automatically. You must know your login password in order to log in. If you’ve forgotten the password, you will need to reset it before you begin.
Test while in safe mode. Same problem?
After testing, restart as usual (not in safe mode) and verify that you still have the problem. Post the results of Steps 1 and 2.