1) In order for Windows to see a HDD on the "My Computer" screen, a partition needs to be present on the HDD.
Another way to view all of your storage devices in Windows, is to access the Disk Management screen. You can do this via Control Panel, or access it directly by using the "run" dialog box like this:
Hold down the WIN key on your keyboard while pressing the r key. You should see the "run" dialog box appear on your screen.
Enter this in the dialog box : diskmgmt.msc
When the DIsk Mgmt screen launches, you should see something similar to this (my Disk Mgmt screen)
Your new SSD drive will probably appear as "unallocated" if you haven't created a partition on the new disk.
2) Windows 7 does have a built-in Backup utility although I've never used it.
I use Acronis (ver 2011) to Clone my C HDD as well as Imaging with "Macrium" (another backup software tool).
If you're interested in performing a complete HDD backup which will boot into Windows and contain all data on the HDD, you can either Clone the HDD or Image (full-HDD Image) the HDD.
I use both methods, using Cloning for my short-term HDD backup routine (every 2 weeks) and periodic full-HDD Imaging to provide multiple HDD recovery paths in the event I run into problems with a Cloned HDD.
If you're wanting to process a complete HDD copy via Cloning or Imaging to your new SSD, you shouldn't need to prep the new SSD, or that's been my experience with Cloning spinner (non-SSD) HDD's for several years.
For example, When I Clone, the Target (sometimes referred to the "Destination" HDD) is nearly identical to my Source HDD since I had Cloned the Source HDD 2 weeks earlier, if I'm following my schedule.
I use "Diskpart" to delete all partitions, including the partition table, so that when I've launched Acronis (or another software backup tool), the Target HDD will appear as "unallocated". That helps differentiate between my Source & Target HDD's.