All that is going on when you select the curves of interest, is that JMP is just selecting the rows in the data table that represent the curves selected in the graph.
Saving of the graph can be done in several ways. The easiest is to copy it to a JMP Journal. Interactively, this is easily done by selecting the graph to bring it into focus, and then simply do, CNTL/J. This will move the current image of the graph to the journal. Subsequent changes to the original chart will not change the image that was copied to the journal.
An easy way to display the rows one wants to display, is to the row selection built into JMP.
Right click on the header of the RowState
and it will bring up the selections below
If these built in features do not meet your needs then a JMP script will probably have to be written to select the desired rows.
Jim