Why is my mid-2010 macbook pro having trouble booting and crashing?

Question

I have a mid-2010 13 inch macbook pro that has recently began to have severe problems. It started after I shut it down one night and woke up to the sound of my mac's fans running loudly and my computer running extremely hot. Apparently, it must not have finished shutting down that night and something went wrong with it when i closed the lid. Ever since then, the computer will either take a long time to boot up (the apple symbol will take up to a minute to appear, and then up to several more minutes to go to my desktop) or it will display an error (either the folder icon appears instead of the apple logo, or the apple logo appears and then after a minute or two is replaced by the circle slash). Once i'm in, the computer seems to function normally, but anytime it needs to access the disc there might be another delay, or the computer will freeze, forcing me to shut it down again. The most common time this occurs is when I am waking the mac from sleep. In addition, the cpu would run hot (~85) which would imply i was doing high cpu activity, except i was usually doing very little, activity monitor showed low levels with sudden intense spikes whenever i did something.
I've run several diagnostics to try and determine the problem, but none identified an error:
booting computer from different computer/startup disk - runs fine (apparently. gave my mac to someone else to try this test, could not see results for myself)
Thorough Apple hardware test: no problems found
disc utility: disc/discpermissions verified and repaired with no issue.
PRAM/VRAM reset: actually sped up the lagging of the system once it reached the desktop and intense cpu thing but booting problems still persisted.
archive and install: some of the time it would fail to install in the middle of the installation, forcing me to start again, but once i completed it, issue still persisted.
clean wipe of drive with restoration from time machine: computer speed better but still trouble when booting.
Clean wipe of drive without migration of data: computer running and booting quickly, but occasionally still received circle slash instead of apple logo.
At this point, i don't know where the source of the problem lies. The computer is old, but has never had an issue until recently. The problem is inconsistent, and all testing implies it doesn't exist, which makes its diagnosis extremely difficult. As it is a 2010 mac, it is clearly out of apple care and warranty so i cant go to a genius. I have a weak battery (80%) and my screen lid is a little loose from frequent use, but those are the only blatant problems i'm aware of with my mac.
Any suggestions? I can gladly provide more info as needed.

Answer 1

When you have the problem, note the exact time: hour, minute, second.  
These instructions must be carried out as an administrator. If you have only one user account, you are the administrator.
Launch the Console application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad. Click Utilities, then Console in the icon grid.
The title of the Console window should be All Messages. If it isn't, select
          SYSTEM LOG QUERIES ▹ All Messages
from the log list on the left. If you don't see that list, select
          View ▹ Show Log List
from the menu bar at the top of the screen.
Each message in the log begins with the date and time when it was entered. Scroll back to the time you noted above.
Select the messages entered from then until the end of the episode, or until they start to repeat, whichever comes first.
Copy the messages to the Clipboard by pressing the key combination command-C. Paste into a reply to this message by pressing command-V.
The log contains a vast amount of information, almost all of it useless for solving any particular problem. When posting a log extract, be selective. A few dozen lines are almost always more than enough.
Please don't indiscriminately dump thousands of lines from the log into this discussion.
Please don't post screenshots of log messages—post the text.
Some private information, such as your name, may appear in the log. Anonymize before posting.