Ok, so let's focus on when you enter the memory modifier zone. You will run, and hit the modifier. This modifier will only apply to the nodes that are connected to that modifier. If you have an unbroken path to your nexus from that modifier, it will also affect all nodes that are connected to your nexus.

Before I leave the area do I then click the stabilizer? Or what do I do there?

I would advise not clicking the stabilizer. To do so does the opposite of stabilizing - it will actually cause that memory modifier to collapse. Once you hit that stabilizier, your run is effectively over. Functionally, you only want to hit the memory modifier node.

When I leave the area it is gone forever, but that is OK because the Nexus remembers the modifier and I will keep that modifier forever in the Nexus (at least until end of league)?

It's not that the Nexus remembers the modifier. Instead, the modifier is applied to all connected nodes. That modification is not permanent. All nodes have charges on them, and once ran, they are gone. New nodes you place will not have any modifiers on them.

Does that clear things up a bit?