Can I grab a section on the timeline and copy it and put it in its own event? or is there a better way to do it?
No. You cannot copy a full clip or a segment of a clip from the timeline into an event or into the Project Media browser. You can drag a full, unedited clip, from the Project Media browser view into an Event or into the timeline. You can drag a full, unedited, clip from an Event into the Project Media or into the timeline. Events can only hold unedited clips, not segments or clips with edits, so if you want to save an edited clip or a clip segment from the timeline you need to copy and paste it into a newly created project that will act as a sort of library or archive for unedited clips that you may want to put into it. The edits will carry over.
When you create segments of clips on the timeline each segment, while it displays as a segment, contains the entire clip. So, you can drag out the edges of a segment and restore the entire clip. When you create a segment it does not display as a new clip in the Project Media browser view, because the full clip is already in the browser, and the segment of the clip is only an edited version of the full clip. Edited versions will not be displayed in Events or in the Project Media view.
Fulls clips that are stored in an Event are accessible to all projects and can be dragged into any project.
As for labeling clips in the timeline, that cannot be done other than by overlaying a title onto them.
-- Rich