I have no new ideas about the logging out of students on your site. I don't think it is to do with the MySQL version. More likely it is caused by the authentication system that your Moodle is using.
I understod that this issue is not confined to TaskChain activities on your site. It happens when viewing other activities and resources too right? Furthermore, I understood that your Moodle uses something other than Moodle's own username+password for authentication. That is where the cause of the issue lies I think.
I have not experienced it on my Moodle sites, and I have not heard of similar behavior on any other Moodle site except for yours.
Regarding the navigation menu, I think the the behavior you describe is the normal behavior for the Moodle navigation block. It shows all the activities in the current course of the current section.
For example. the attached screenshot shows what I can see as a student on one of my courses that has a lot of TaskChain activities. I can see the current TaskChain, "1. Vocabulary: Jumbled Letters", and it has a link for the "My attempts" report. There are also links to all the other TaskChains in the same section of the course as the current TaskChain.
All the other TaskChains in the course are not currently visible on this menu, but if I expanded the other sections, but clicking on the expansion icon, then the other TaskChains would also become visible.
If you have very many TaskChains in one section, it will make the navigation menu very long. I suggest you consider reorganizing your course sections so they each contain no more than 10 activities.