I need help re-installing my OS on a new hard drive. Having MAJOR issues with this simple task.

Question

I have a MacBook Pro 15" (mid 2009) I recently needed to get a new hard drive for it, the old one stopped working. It came with OS X Snow Leopard already installed from Apple, and it also came with the install disks. I have booted up using the install disks and put OS X Snow Leopard back on the computer.
I tried booting with a USB install disk of OS X Lion (10.7) but had no luck on multiple occasions. It would only get through the initial step of the install and then say it couldn't download additional components needed for OS X 
So finally, I decided to try a full erase and reformat of my brand new hard drive (less than a week old) and stated with the basic software that came with the computer. Installing that worked fine, until I tried to do a software update through the built in program. My computer refused to download 10.6.8 from the apple servers, so I had to find it on a 3rd party website. Once I found it and downloaded it, I finally got it installed. There were 12 more updates after I did 10.6.8 and none of them worked when I tried downloading and installing as a group. I had to do each one individually, the last one being a security update.
Up to this point, the Mac App Store didn't work (even after the 10.6.8 update), I couldn't download any of my previous purchases. I needed to download lion, to see if having it on the Mac hard drive itself would help with the install, but I couldn't do that until after the security update.
Other than that, the Mac worked normally except for during each start up, the apple logo would flash along with the forbidden sign and the folder with a question mark on it. It flashed for a few seconds randomly, then would boot up normally or say that my computer would need to be restarted.i figured this issue would be fixed after I successfully installed lion.
FINALLY I got the App Store to download lion, and began the install process. Halfway through the install, it failed and when I tried to restart the computer, it continuously flashes through those three symbols at startup. I can't get it to do anything else. I'm so ****** off I'm ready to just sell it for parts.
My questions are:
      1. Can I install OS X Lion OR OS X Mountain Lion clean from the start after a full format of the hard drive?
      2. How do I go about installing clean and get my computer to stop flashing those symbols during start up?
*I have performed the Apple Hardware extended test and all systems are reporting normal function*

Answer 1

Okay, so, fast forward from July to now -
- I've visited my closest Apple store's genius bar twice now. (Over an hour and a half away.)
- My computer spent at least 8 hours behind their closed doors in the workshop
- Several complete erases of the harddrive have taken place
- Most of the Apple geniuses I dealt with were nice and very knowledgeable, but they were and still aren't 100% sure exactly what is wrong with my computer.
The good news is:
- They were able to at least get it running again, with a clean install from their servers of OS X Lion 10.7.5
- They believe that my problems stem from either 1 of 2 things:
        - Faulty Hard Drive (even though it is brand new)
         - The wrong build of software being installed.
Now, while the wrong build does seem like it could be the problem, it doesn't make sense in the aspect that all of the initial problems began when the first hard drive failed. The same problems continued when I changed out the failed hard drive with the brand spanking new one, and only slightly improved when I was able to successfully install the OS that came with the original disks for my computer. I'm not convinced this thing is back to normal, but hopefully I can make it last through getting a shiny new retina Macbook Pro in the coming future. Thank you so much to everyone who read and replied to this post!
- Andrew =)