I’d just like to know if using vector graphics (specifically .swf format) is possible with Slate. My previous experience with them encourages me to use them for my UI needs. I know this is a simple question (hopefully), but I haven’t found a thread about this yet, and I am rather unfamiliar with Slate until very recently.
With Slate, not unless you wrote a widget capable of rendering said content.
However, depending on your financial situation you could look at Autodesk Scaleform. http://gameware.autodesk.com/scaleform
Alternatively, you could likely do something with HTML5/CSS using Coherent UI, Radiant UI or Vaquole.