Hi srkmagnus, thanks for the recommendation.
I've cleared the cache (or at least attempted). Here is the strange part-- the phone claims that the 300-something MB of cached data was eliminated down to about 3 MB, but the phone still acts like that data is still there. Shouldn't I have had to relog back into apps that normally keep track of your username and password? Facebook (for example) just went right back to my account with no login required.
Some changes have happened related the SMS as well, but it not at all fixed. After clearing the cache, I started receiving a few texts from a friend that I know were sent almost two weeks ago. Just two texts came in (from two different people). Then I attempted to send a text to a family member (brother) who was nearby. We clocked the transmit time at 8 minutes for a "Test" message. Then I send another one, with a 15 minute delay. This is a drastic improvement upon the 8-500 hour delay for texts sent prior to clearing the cache, however I am still not satisfied with the result. The "Test" message was received on his phone, but my phone claims to have not sent it following a reboot.
I am still quite sure that the phone will be fixed by a factory reset, so really my goal is to keep my old contacts and pictures. Unfortunately, I cannot access the phones hard drive from any computer. The phone does not register as a storage device.
The real source of confusion with this bug is the breadth of error. SMS, Google Play/Apps, and data storage overall do not intuitively have anything in common. Apps that require access to multiple hard drives are the ones that fail though. My hypothesis is that the phone cannot draw information from more than one data storage location at a time into the RAM. This is supported strongly by how the SMS behaves during power cycling.
When I power off and on again, SMS performs an action that appeared frozen before. Now this is fascinating. I just tested this while writing out this extended bug report. Sending a message on SMS to my brother does not work unless I follow these steps: send message, power off, power on. The moment that the radio turns back on after a power cycle, the message is received. At exactly that moment (that the text is received), my phone says that a message sent previously to the most recent one could not send. The phone has not been able to receive a confirmation from the tower of the text being sent. It is also unable to receive a message from another phone. I believe that the phone stores the message in its own memory before sending, then freezes on the "send command". The send command proceeds when the phone is rebooted. There is a storage of the text into the phone memory, then an access to the SIM card to send to the tower. I believe the phone freezes while attempting to access the SIM card.
Things I can't explain yet. Apps that mostly work. Facebook works just fine, except that I can't open links. OkCupid works without a hitch (which is odd, because I believe it uses GPS). Snapchat works as well as it always has (that app is particularly buggy). Skype works. Dropbox works... wait a second.
PARTIAL SOLUTION: Dropbox can save all my photos. Download Dropbox and start a picture upload.
This has been a text-based live stream of debugging by fouldsc. More to come later.