How can I get my Mac to load past the start-up screen?

Question

My MacBook Pro (17" Mid-2009 model) that originally came with Snow Leopoard was acting up recently. Currently, it was running OS X Yosemite. I had been using the computer and then left it in sleep mode for two days while I was away. When I came back, the computer was turned off. At first, I thought it had simply run out of battery power but a quick press of the battery-level button on the side showed it had around 80% of battery left. I did not turn it off before I left, so I assumed another person had turned it off for me.
I proceeded to turn on the computer, but it started to make a faint, but abnormal noise. It was not coming from the disk drive, more on the lower left corner of the computer. It was repeating every few seconds. It then completed start up and I was presented with the log-in screen. The normal icons for each profile showed, but the icons for the menu bar at the top and the "sleep", "restart", and "shut down" along with other normal computer icons were completely weird. The closest I could describe them as was a more pixelated version of the infamous "missingno" from early Pokemon games: garbled pictures of once-normal icons.
The the comouter was acting much slower than usual, but I was able to click on my profile to enter my password. The animations were sluggish and distorted. It didnt even show that I had entered any characters, but I knew it was attempting to log in because the placement of icons were arranged to where they would be in a login. After a few minutes, I tried to shut the computer down by pressing the on-screen power button, but to no success. I forced the comouter off by holding the power button on my Mac, where is sputtered down with a flash of the screen.
I waited a few minutes, and completed a NVRAM restart. The sound came on along with the screen, but the loading bar would get to only 1/3 of the way before sitting for 3 minutes then dissapearing as the computer shut off. I attempted to restart several times, all met with the same failure. At this point, I had tried a few things suggested on websites such as entering single-user mode and entering "fsck -fy". I was unable to enter safe mode or disk utility.
Then like magic the computer actually completed start up and I was able to log in. Unfortuantely, I was only on for about 10 minutes before the comouter shut down yet again. Through the next few hours, I was desperately trying to restart the computer, only being able to enter disk utility once. The disk utility said there was nothing wrong with "Macintosh HD" when I "Verify Disk"-ed  but that something else was wrong when I clicked on "Repair Disk". (I don't even know if this is even the right kind of thing to do in this type of problem, but I tried it anyway.)
So, I have tried different things like removing my external mouse and re-installing OS X (Which I could not do because it only gave me the option of selecting "Recovery Drive" because the normal "Macintosh HD" did not appear). But nothing has worked so far. From what I have seen, this also doesn't look like an issue with a graphics or video card because the screen is no longer glitchy (I read a discussion about failing NVidia cards causing a lawsuit and free replacements but they were a few years before my Mac was purchased) but I do not know what a problem with a logic board would look like, nor a failing hard drive. From what this looks like, I think I am going to have to fork over hundreds of dollars at the Apple Store just to fix something that is wrong. Hopefully, I won't lose any data.
Here is more information about my computer:

Answer 1

Slow, intermittent.  Sometimes bootable sometime not. Odd behavior.
Sounds to me like a classic case of your HD failing.
Your SMART status may still show verified, with no errors on the Apple Hardware Test, Disk Utility shows no problems on Disk Repair-  this is of no help.
I trust you have a good back-up and a bootable clone.
http://eshop.macsales.com/shop/hard-drives/2.5-Notebook/