Mac not booting/running slow as molasses

Question

Alright. This is going to be a large and multi-scope question, so please bear with me, everyone.
Here are the (general) specifications of my MBP:
Mid-2012 15.4" MacBook Pro
500GB HDD, 4GB of RAM (I believe)
OS X 10.9 (Mavericks)
Now, for the questions. The two are chronologically related; I believe also in other ways related as well. In order of what happened:
1. My machine (what I'm typing on now; I'll get to that later) has been running slow as molasses for the past few days. We're not talking 'not as fast as when I bought it' slow, we're talking 'slow as a fully-loaded mule going uphill' slow. Things started to occur the other day, when, for the second time in this computer's existence, it was dropped. You don't need to educate me on why I shouldn't drop a computer; I know what can occur and I've accidentally dropped many other devices before, and had to suffer the consequences. The first occasion was when a friend of mine as using the machine and dropped it while walking with it. The screen was spiderwebbed (LCD fine; display glass has spiderweb cracks across it), but other than that, no real issues. The second time, about a week ago, was when it slipped off my lap onto a tile floor (not that laminate stuff, the hard grouted tile). It was in a case and the display glass only suffers minor advancement of damage (barely noticeable...still like that...as a student and working to support myself as a programmer, I can neither afford monetarily nor the downtime of having the computer sent to Apple for repair, especially for something that doesn't affect the functionality of the machine).
The next part isn't necessary to the problem description, but bears saying - I am not a novice shouting 'my computer is slow, plz help' without any description; but I am not the best user on the face of the earth. Most of my experience in my life has been on the Windows platform; the OS X operating system handles similarly enough in terms of general functions but has enough internal difference (I know this is a major understatement - they are two fundamentally different OSs) to still cause me problems when trying to perform maintenance or "power-user" tasks.
Now, back to the actual problem, the computer immediately turned off, which I found to be immediately suspicious. Upon rebooting, the computer successfully turns on and functions without issue, but is as the title and first few sentences describe, is slow as all ****. I have recently done two things to this computer (major things) which I feel could contribute to the issue, namely:
Partitioned the HDD to install Fedora 19 - I'm in the process of working on an open-source project involving an IBM SDK which requires Fedora - as I live during the week in the Bay Area and commute to and from home, I only have intermittent access to a desktop which I would otherwise use for it. The computer's speed of operating didn't seem to bat an eye at this process.
Installed two things, namely the upgrade to OS X 10.9 and Photoshop CS6, which I use to generate the GUIs for my applications (iOS development primarily).
Both of these didn't cause any issues with the speed, but after the drop, this thing handles like it has Alzheimer's. Now, I know that dropping the computer isn't a good thing; if anyone can elucidate me further to how such a thing could be detrimental I'll be the first (and only one) accepting full responsibility for it. Mostly though, it's just been an annoyance of little enough significance in terms of productivity that I could still work without it.
Which brings me to point number two:
So, in an effort to generally bring up the speed, I noticed that Fedora had created two partitions, one called 'disk0s5' or something of the like, and one called 'untitled', taking up about 30GB. I know that space can be an issue due to operating systems handling by copying data around, shuffling it, etc. but as I still have over 400GB free, I only did this as a test.
I removed the partitions (did not touch the one titled Macintosh HD...we all know what that leads to) and rebooted, which brings me to present time & day.
Might as well list this to establish a chronology:
I start the restart process, and head to the bathroom in the meantime.
I hear the startup chime.
Upon returning, the computer is off, so I know something is up, but I manually boot it to see what.
Upon the white initial boot screen, it loads a progress bar, much like the Mavericks install.
The typical spinning throbber loads and the load bar progresses to about a tenth (takes 15-20 seconds) and the computer shuts off.
Now, for me, this is bad mojo. I haven't done a full system backup on this machine (I know it's bad form, no need to educate me on this, call it the 'not wearing a seatbelt' for computing), but I know that ideally, the partition should be intact, so I intend to clone the drive to an image and back it to disk or portable HDD before reinstalling 10.9 or sending in to Apple.
For the obvious... I am on the computer and it isn't booting as of now, but I was able to get into the recovery partition via option-boot and use Safari.
Please advise. Again, I'm not a novice, but Mac OS X isn't my primary forte so you will need to break things down somewhat for me. I will provide anyone with more information upon request, if need be, limited to what I can access in the partition.
Thanks in advance,
Eli Fedele

Answer 1

Boot from an external system disk:
Format the external disk Mac OS Extended (journaled) running Disk Utility form the Recovery Partition.
(Boot then hold down Command and R simultaneously).
Install OS X onto that external disk.
Boot from that external disk (Boot then hold down Option).
Using the Finder drag as many folders as you can to the external disk.
As a last-ditch effort you can use recovery software to recover files the Finder will not copy.
DiskWarrior:  http://www.alsoft.com/diskwarrior/
uFlysoft          http://www.uflysoft.com/
McAfee Data Recovery for Mac:  http://mcaf.ee/x0pfi