Is there a way of knowing the beginning and ending timecode of a particular scene on the timeline?

It's displayed in the project panel for each clip, or in the source monitor is you just click on the scene in the timeline -- it'll show you both the actual beginning and end points of the clip as well as the in and out points.

Also, you could drag the timecode effect onto the clip and have the TC actually display in the frame, then remove it once you no longer needed it.