Can't boot up/need to save files

Hi everyone,
My Macbook (Aluminum Unibody-before they labeled it "the pro") started slowing down when I installed snow leopard recently. It was jamming up a lot and I would have to restart often. On my most recent lock up, I restarted and it never went past the apple boot screen. I booted from the system DVD and ran disk utility, to no avail. There were all sorts of errors. Brought the computer into best buy and they tested the health of my hard drive-it is in perfect condition. THey suggested I just reinstall the OS and everything on my computer would be saved. I went to do this and received this error "You cannot install Mac OS X on this volume without changing your installation settings. Click Options to change the settings. The contents of this volume cannot be changed"
When I click options, I receive this:
"The destination disk must be erased for installation.

The detination volume will be completely erased and a new compy of Mac OS X will be installed on the volume. All your data will be lost" Then gives me these options "Format as: Mac OS Extended (Journaled) or Mac OS Extended (Case-sensitive, Journaled).

You gotta talk to me like I'm someone with no experience with this sort of thing. Because.. I'm not experienced with these issues. I did all the startups that are suggested... the fsck (or whatever) thing (doesn't fix anything), the single-user start up, start up in safe mode (doesn't work).

Now, I can boot up with my system disk and reach the disk utlity. There is an option there to "restore" by dragging the Mac HD to the top field and the external hard drive to the bottom field. It does say it will write over any other files, however.

Here's my issue-first, I want to make sure that it will really make a full copy of everything, just as Time Machine does. The only hard drive I have that is WORKING with this, is the one that my time machine backups are on. I'm assuming all of my backups (there are many), will be erased and written over. Is this so? And will backing up by this method, really work to where I will be able to access all my files? Will I be able to restore from this backup after the OS is reinstalled? I hadn't backed up since December 3 and there is some important information and files on there that I just can't lose. I already tried switching out the hard drive from my computer to another mac, but for some reason, the screw keeping the hard drive attached, is stripped and there's no way to get it out. Very strange since I've never tried.

Any help would be greatly appreciated. Thanks so much!

Thank you!

 

Will this perform a full system backup?

I posted just before but figured I'd make this an easier yes or no question.

Because my computer won't boot, but will boot from the system CD, I'm wondering if i go to the "restore" tab and bring the mac HD to the top field and an external hard drive to the bottom, a GOOD copy of the entire hard drive will be made. One that I can then restore my computer with, after I reinstall the operating system. I'm wary about giving it a try, because all of my time machine backups are on this external hard drive (yet it's the only external hard drive it wants to work on)and I'm afraid they are going to be written over and then I'll end up with *nothing* ( I want a more recently copy of my hard drive). Best Buy tested the mac for me and the hard drive is in working order- there are just a bunch of errors that aren't allowing the computer to start.