Computer keeps getting folder with a question mark (?) I changed the HD and the SATA cable, now what?

Question

The problem started when my computer was having problems reading the HD by slowing down and freezing. The Apple Genius told me the computer needed a new HD, and that it would cost $228 to replace it through them. I opted out and purchased a Hd myself and installed it. Everything was good for about two months until one day the screen went black and I smelled something burning on my Mac, yes like fire. Upon inspection I found the burning smell came from the middle of the SATA cable and that it shorted. I ordered a new cable from Hong Kong that took 22 days to get here. It still does not read the HD. I tried a SATA cable from a 2011 MacBook that still works (had to stretch it a little but still connected) I also tried the previous HD that was in there and two other HD from 2 different MacBooks that are still functional. Nothing worked. Is my motherboard toast as a result of the SATA cable dying? The computer reads an external drive, however, and can boot into time machine. The cable, btw had been receiving physical damage from the case when the MacBook lost its little black feet. This is where it fried as the section that had marks from the case melted the plastic right underneath.

Answer 1

Prep your new drive:
Drive Preparation
1. Boot from your OS X Installer Disc. After the installer loads select your language and click on the Continue button.  When the menu bar appears select Disk Utility from the Utilities menu.
2. After DU loads select your hard drive (this is the entry with the mfgr.'s ID and size) from the left side list. Note the SMART status of the drive in DU's status area.  If it does not say "Verified" then the drive is failing or has failed and will need replacing.  SMART info will not be reported  on external drives. Otherwise, click on the Partition tab in the DU main window.
3. Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID (for Intel Macs) or APM (for PPC Macs,) then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Partition button and wait until the process has completed.
4. Select the volume you just created (this is the sub-entry under the drive entry) from the left side list. Click on the Erase tab in the DU main window.
5. Set the format type to Mac OS Extended (Journaled.) Click on the Security button, check the button for Zero Data and click on OK to return to the Erase window.
6. Click on the Erase button. The format process can take up to several hours depending upon the drive size.
After formatting has completed quit DU and return to the installer. Install OS X.