Thanks for posting your question. I believe that this type of graphic falls under the category of Charts and Graphic Organizers (Section 7.7 of Guidelines and Standards for Tactile Graphics). 188.8.131.52 says "Every attempt must be made to place an organizational chart on one tactile page. If this is not possible, a fold-out page, which is preferable, or facing pages can be used." It then refers to Appendix D, where in D.3.5.1 it describes how to create a fold-out page.
I realize that this tree structure is not the same as an organizational chart, but I believe that you should follow the same principles. For a tress with multiple columns, the headings could be keyed to conserve space. And although I cannot point you to a specific rule, yes, I do treat the columns to the right as table headings above a braille separation line. Even though treating this partially like a table, it is not necessary to use rows of boxing lines across a tactile graphic. If the column headings are keyed, it is not likely that you will need to use guide dots between the columns, but in some cases you may need to. Do not, however, use guide dots on the left side of the tree structure where the lines are shown, use them only as necessary for the braille columns of information.
I have never seen the use of a key number indicating a continuation on the next page, apart from where it is used in Chemistry Code to illustrate the structure of bonds. It would certainly not be my first choice for this type of diagram.
The lines themselves can be lengthened or shortened as needed, and can be "spread open" wider to allow them to align with the braille text. It seems to me that the example you provided at the top of your attached print page, should fit nicely on one page (if the headings are keyed). For more extensive diagrams I would recommend that you use a fold-out page.
I hope this satisfactorily answers your questions.