Yes, your info makes sense. I can suggest two further routes:
1. Try connecting the CD and your HDD to the other two SATA ports, see if that makes any difference. Check in BIOS Setup that all the ports are Enabled.
2. See if you can find a friend who can let you test your HDD on their machine. All you need to be able to do is connect it to an unused SATA port and see whether their BIOS detects the HDD properly. It should be able to show you a maker name, a model number, and a size at least. If it can't, that would indicate a faulty HDD and you could go back to Dell Tech Support with that info.
I note one missing piece of info. This machine, according to the info I looked up on CNET, came with Win 7 Home Premium plus a bunch of other software installed on its original HDD. Your post makes it sound like your machine now has only the CD drive and the new HDD in it. So, where is the OS installed? Was this HDD supposed to have an OS and software pre-installed when you bought it? Or, are you expected to Install an OS and your own software on the new HDD.
Of course you cannot boot from a blank new HDD until an OS is installed on it. Now, I grant you, the error messages you are getting do not sound like the normal errors for a boot device that is working and just does not include an OS to boot from. They indicate that the boot device you've tried to use just is not functioning at all. What is odd is that the BIOS says the HDD is "installed" on a SATA port, but not "detected".