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Mac won't start up - files not backed up - Genieo behind problem?

Time:October 11
Hey guys,
I'll give you some background information on the matter so far; I bought my iMac 21.5in last year. It's currently running OS X Mountain Lion. I've had this problem where whenever my mac freezes (which has been quite often over the past two months) and it's still frozen after 10 minutes, I just turn it off manually. Then when I switch it back on, a kernel or folder with a question mark on a white screen of startup comes back to haunt me. Usually when this happens, I just unplug the power point and switch it back on after a few hours or the day after, which seems to fix the problem. Well that's not working this time.
Yesterday I tried updating my flash player and I clicked a link which I thought was the official adobe site and pressed download. Upon downloading, it was standard procedure, until the Genieo icon popped up on my desktop. I've encountered Genieo once before (around a month ago) and it froze my computer. When I was able to start it up again (unplugging power point and turning on computer after a few hours) I was able to get rid of it by deleting everything Genieo related. Well, not this time. When I first saw the icon I quickly put it in the trash and emptied it, but it was too late. It had already put the "InstallMac" as all my browser home pages. My computer then froze and I had to manually switch it off.
Now, when I start up my computer it won't make the start up noise anymore. I can't get past the the folder with the question mark on normal start up. Here's what I've tried so far and what happens:
- Holding "Shift" to start up "Safe Mode". Folder with question mark comes up and nothing happens.
- Holding "Option" for "Start Up Manager". I click my volume and the kernel comes up.
- Holding down "Command + R" for "Recovery Mode". This works and the globe comes up on my screen and then takes me to the "Recovery Mode" menu.
So we've established the only thing I can do upon start up is go into "Recovery Mode" (unless there are any other startup key combinations that will assist my problem that I don't know about). The options it provides me with are:
- Restore from Time Machine backup
- Reinstall OS X
- Get help online
- Disk Utility
Because I'm an idiot and thought nothing would ever happen to my iMac, I have never backed up my files onto an external hard drive, so Time Machine isn't an option.
I've tried reinstalling OS X. When I click here it gives me the option to reinstall Mountain Lion OS X. I click on it and it comes up with a pop up along the lines of this is already installed on this volume. One time I clicked on Reinstall OS X and Mavericks came up instead of Mountain Lion. This only happened once and I'm not sure why. I clicked on install and waited for it to install Mavericks OS X for like 30 minutes, but then came up with an error telling me it was unsuccessful. I've also tried installing OS X onto an external hard drive but it failed. So that isn't an option.
Now there's only Disk Utility left. The "First Aid" tab gives me the option to verify and repair my disk.
Now, the majority of the times, I verify my disk and it tells me that there's nothing wrong with it, so I restart my computer and the folder with the question mark or kernel come up upon start up.
Only a few times have I encountered where I verify my disk and it tells me to repair it. Then when I repair it tells me it was unsuccessful and to back up all my files. But I haven't got my files backed up.
I've tried restoring my disk onto an external hard drive through the "Restore tab" but it always fails.
I'm a noob when it comes to mac, so here I am. The thing is, the kernel/folder with question mark on start up screen has happened many times before without Genieo, just whenever it freezes or whenever something goes wrong. But this time with Genieo I can't get past it.
If any of you guys could help me that'd be sweet. If I have no luck I'll have to take it to the apple store, which is really a last resort, and something I don't want to do.
Thanks heaps,
P.S. I made a post before this regarding the same matter, but decided to make it again with a bit more information. On that last post Genieo_support commented on it acting innocent saying Genieo wouldn't be behind this, and have also provided an uninstall link, which I can probably get to through the "Get help online" option on "Recovery Mode". I don't trust Genieo as it is already, but is this link actually an uninstall link?
That is a great site and provides some good information on removing the adware Genieo. He has posted some information on removing Genieo when not being able to get onto my computer, which is exactly my situation. Under this heading he has put a terminal command to remove the Genieo file. He has also written:
After you have executed the “rm” command successfully, you should be able to restart the computer successfully.  If you get an error saying “No such file or directory,” you have made an error somewhere in the file path and will need to correct it. If you cannot figure out how to correct it, seek professional assistance.
Unfortunately for me, I got the "No such file or directory" error, so I don't think there are any other ways to fix my problem.
I've checked many sites on removing Genieo, fixing the kernel and fixing the folder with the question mark, but I seem to have an extreme problem that the solutions can't resolve. My only option now is to seek assistance for the apple store.