Not only would Home Premium work, you wouldn't be able to find Home Basic if you tried. (In fact, you can upgrade to any Windows 7 version ... except Home Basic.) The trick is to use Windows Anytime Upgrade, which is built into Windows 7. This will let you upgrade very quickly (10-20 minutes) and do so electronically without the next for a Setup DVD. For more info about Windows Anytime Upgrade, please read Paul Thurrott's  feature focus article.

Reinstalling Windows on the Same PC
I use Windows 7 Home Premium installed with an upgrade from Vista Home Premium provided by Toshiba. I have reinstalled Windows 7 on my laptop many times and there has been no problem with activation. Is it because I am reinstalling my copy of Windows 7 on my laptop what was previously tied to the key given to me that it is not counted as moving of license?

Yes, that's correct. You can reinstall Windows to the same PC an unlimited number of times using the same product key

What is Windows Aero?
How does the Aero UI work? I know it does something with video card RAM and it's written in XAML but just how does it work? Can you explain how it works to a layman?

Basically, it's just Microsoft's version of a hardware-accelerated graphics-based user interface, where the GPU is used to render the 2D desktop instead of the CPU. Microsoft calls this technology "desktop composition" and Aero is the brand name for the feature, which is really the "Desktop Window Manager," or DWM. The primary advantages are performance and graphical fidelity (i.e. a better looking screen that can take advantage of GPU effects like transparency/translucency).  Here's a Microsoft technical document about this topic.

Why Are There Two Program Files Folders in Windows 7?
I installed Windows 7 (64-bit Professional) and noticed there were 2 Program File folders (Program Files and Program Files x86)?

64-bit versions of Windows Vista and 7 maintain two different Program Files folder structures, one ("Program Files (x86)") for 32-bit applications and one ("Program Files") for 64-bit applications. You will only see a single Program Files folder on 32-bit versions of either OS. This is by design and is nothing you need to think/worry about.

Windows.Old folder:  I noticed today that my main OS hard drive (a 74GB Raptor) was 6 GB shy of being full. I was shocked to see this, because on this drive I only install the OS, my anti-virus software, and things like Firefox and other critical updates (like drivers). Disk Pace Fan shows that there are three folders Names Windows.old, Windows.old000, and Windows.old0001 that are taking up nearly 60 GB of my drive. What are the Windows.old folders doing that they are taking up so much space on my hard drive?

If you perform a Custom Install of Windows on a hard drive where you already have an installed version of Windows, Setup will create a Windows.Old folder structure that contains a backup of various folders and files associated with the previous Windows install. If you reinstall Windows several times in this fashion on the same hard drive, you may see additional Windows.Old folders, as you are seeing.

You can delete these folder structures and reclaim the lost space. But you may want to make sure there's nothing in there that might want first. Microsoft has a number of resources that are relevant here:

Phone: 210.422.8049
Email: helpme