Background Border Settings

When I am setting up background border settings, is this limited to square corners only? Is there a way to allow the user to add a border to a circle or oval?

The BG border setting is a pretty old feature, and for this it is very limited.
So sorry, but cannot be used with other shapes but rect.
Assuming you want users to be able to control things like thickness and color, you can create it as a shape in designer and save it as a template. Otherwise, just use a single color clipart

Is there a way to adjust the size of a background color? Let's say I have a sign that is 12" x 18" that I require a 1/2" border of white all the way around, I would like the customer be able to choose a background color, let's say black, it would then show up on the screen would show a 1/2" white border all the way around. Is this doable?

The BGcolor, by definition, is the BGcolor of everything. It is just a full layer on bottom of everything so it really has no "size" but is always the full size of the product.
That said, to do what you want you have mostly 2 ways:

1) if you want user to be able to define BOTH colors, then the only way is to use 1-color cliparts in a template and allow user to change only the color of those

2) if the "white" is always going to be "white", and so if you really want users to select the other color, you can use regular BGcolors and play with size margins (which determines the size of the product as well as the size of the BG) or make proper use of BG/FG masks for the product to visually "create" the effect.
If this is going to be your way depends on if you want those 2 colors to be a reference for user only or also meant to be printed, because BG/FG masks are only visual in the designer but are normally NOT exported in the hi-res file.
If you;d like to understand more about BG/FG images, in past Mark did this nice video to give a clear view on how they can be used:
https://www.youtube.com/watch?v=U_E5yZe7pEQ

Thanks! I'll look into the 2nd option.