Are all connected/common nodes located on the endpoints of a way. In other words, ways are made up of multiple nodes in a designated order. If two highways share a node, I assume that the highways are connected and can be traversed. Should I also assume that all of these connected nodes occur either at the beginning or end or a way and never in the middle?
You cannot assume that connections are only possible for end nodes. A road will almost never be split just because some minor way connects to it, and junctions are often modelled using intersecting ways with a shared node that neither way starts or ends with.