In the edge-triangle intersection check, is there any value to knowing how much the two triangles intersected? Or can we also harvest these potential benefits using a more fine-grained mesh?

I don't think there is any way for two triangles to intersect without an edge of one intersecting with the other. For an intersection to occur, there must be some points shared by both triangles by definition. From a barycentric perspective, these shared points must be linear combinations of the vertices of both triangles. From this it is visually clear that there must be some point along the edge of the triangle at which the intersection begins.

Do we ever care about the line segment where the intersection occurs, or are we only interested in the yes/no answer?

Would edge sharing or surface sharing be considered intersection?

Would Minkowski sum help here to find intersection?

If we represent triangles with prisms, how do we tell the difference between collisions and when the paths of the triangles cross, but they don't actually collide because they're not at the same place at the same time?

There are different cases here. It can either be two rays from the same triangle intersects the other triangle (on the left), or one ray from each triangle intersects the other triangle. Cannot think about if there're other cases.

What if motion in time isn't linear? Do our algorithms get much more complicated to check for intersection?