Is it possible to add a text line for each clip, which will appear on the screen whenever that clip is played?

In Audacity you can use “labels” http://manual.audacityteam.org/o/man/label_tracks.html
Labels only work in Audacity projects - they are not supported by other programs.

See also the “karaoke” feature: http://manual.audacityteam.org/o/man/view_menu.html#karaoke

Thanks, Steve. Hillel