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I like to better understand why in the following image I cannot fill the gap by adding a new segment or joining two adjacent nodes on each side. Basically after joining the gap remains. The only time the gap is filled is when joining all four corners, but then they all become a single node (pointing arrow) destroying the concept. None of the four corners are open nodes. Why is this happening and how to fix it?
PS: These are open paths as their end do not meet their start.
I am having difficulty understanding your question to be honest, or what you are trying to achieve. If I've guessed wrong, please forgive me. It's hard to tell much from your screen shot, but it should be possible if these are open paths as you say. You might want to consider adding more details to your question.
It looks to me like you have not selected any nodes. You've only selected the paths.
After selecting the paths with the Edit Paths by Nodes tool (N), hit CTRL+A to select all the nodes. They should turn blue. Then hit the join button. If you just want to select two nodes to join, click and drag around them to select them, then join.
Example
If you want to keep these end nodes and join, then draw an extra line, then select both nodes that overlap and join them.
Example
You are dreaming of joining 2 open paths together in the middle of both paths and expect to get a single new path which has four ends.
4-end open single paths are non-existent in vector graphic programs. An open path has 2 ends and that's that.
Some possible things
You can select your 2 open paths and apply Path > Combine. It doesn't make any joints but make a combined path that can get a path effect like a single path, if path effects are wanted to be inserted. They do not accept groups.
You can break both paths and rejoin the pieces in a new way. Let path 1 have nodes ABCD and path 2 have nodes EFGH. After breaking you have 4 paths AB, CD, EF and GH. You can make paths ABFE and DCGH. Or a single path ABFEHGCD.
If it happens that you for some reason cannot find which is a start node or end node of an open filled path and which are intermediate nodes insert temporarily a stroke.
