I have broke down your OP to bite size pieces to address each point individually
I've settled on starting with the Shuttle SH67H3 Barebones PC.
Not a good idea for a couple of reasons including the following;
1: Your intended use will require a high end video card and these can be large + also generate a lot of heat so you are best to have a larger case which will allow for an ATX size MB and a more efficient cooling system.
2: A larger case will also have room for adding additional HDDs as your video editing library grows, important if you intend to make or have a career in editing/rendering videos.
3: The H67 MB is dated technology and this like numbers 1 & 2 above will limit upgrade options, choose a Z77 platform as a minimum.
CPU - Intel i7 3770K 3.5 GHz Good choice of CPU.
GPU - (undecided) If your budget will allow it consider one of the following GTX670, HD7970 if not a GTX660Ti
RAM - Corsair Vengeance 16GB (4x4GB) CMZ16GX3M4A1600C9B Good brand of Ram but I would suggest the low profile type of Ram be used as this will avoid possible problems that you can encounter with large CPU heatsinks/cooling fans.
HDD - 1.5TB Seagate 2.5" SATA drive w/ 2.5" to 3.5" bay converter I suggest using 1TB HDDs and adding additional ones as you require them in the future.
With my hard drive, it's already been partitioned in half from my previous computer (Macbook Pro) with half Windows 7 and half Mac OSX Mountain Lion. Would either of the OS's run properly upon start-up? And if not, would there be a way to install the hardware drivers onto the hard drive in order for the OS's to run properly?
I am not familiar with Mac OSs and so cannot provide an answer based on personal knowledge but Windows 7 I can give you a definitive answer which is no it will not work, the OS is not configured for the new hardware/has no drivers etc and so will not recognize the components, the other reason why it will not work is because of the licence agreement that you have with Microsoft from when you initially purchased Windows 7 and the product key for it, see my canned text below;
Just a cautionary note, unless your OS disk is the full retail edition you cannot use it with a new MB as an OEM disk is tied to the original MB it was paired with, to use an OEM disk with a new MB is software piracy and therefore illegal.
Exceptions to the above are 1: If your MB is replaced under warranty and 2: If your MB is replaced out of warranty with an alternative type but same brand due to the original model no longer being available, an upgraded MB however will require the purchase of a new OS licence.
If you have a full retail disk and a product key that is not in use on another computer the OEM restriction/s is/are not the same.
I'm also not sure how much RAM I should be investing in. Would I benefit much from 32GB vs 16GB?
16GB is fine for now but it may need to be increased in the future as software evolves and demands more resources.Please note that you should not choose your Ram before the MB has been selected and confirmed as available, this is the same with choosing the CPU + a PSU should be chosen last to allow for all hardware demands
If you provide us with your maximum budget for the build, tell us what if any parts you have that can be re-used such as DVD drive etc and in which country you will be purchasing we can put you a suitable build together which will hopefully also save you some cash in the process.