Hi,
Welcome. Let's see if I can help.
So the goal is to open one file, cut it in different places and then export each individual track into separate files?
The first part is easy with the scissors tool. To save each out individually pretty much requires the merge tool. Click on each then 'save as...'.
It wouldn't be too hard to write code to save out each track as an increasing numbered sequence (1.gpx, 2.gpx, etc). But I'd like to understand the background for what you're doing, if you care to share.
Cheers