Rounding the corners
I started approaching the corners the way I always would: add a border-radius rule to the outermost element. However, when I did this, I noticed that parts of the border, specifically in the corners, were missing. I dug through the layers of elements in the calendar, and found something interesting: in order to avoid doubling up on borders, the FullCalendar library was hiding borders on the top and left of most elements, only showing the bottom and right ones. Then the outermost elements filled in the top and left borders.
I figured this should be something that I can work around, so I set to work on finding the correct combination of elements to apply borders to. After a while of this, and not being able to achieve exactly what I was hoping, I did some digging and came across the outline property. It should look and function just like a border, but it is outside the border, and can be styled independently.
The Solution: I found the outermost elements that were supplying the border, and removed it. Then I added an outline with the exact border style I was looking for. This seemed to work just fine, until I noticed the second challenge.