There are two ways to do this:
Add a ClipBoard Extender app., like ClipMate, and then you can Copy from one Project, Open another, and Paste that material over - you DO need a ClipBoard Extender though.
You can Publish+Share from one Project, then Import that resultant file into another PrE Project. If working with intermediate files (files that will be Published+Shared, then Imported and re-edited), then this article might be useful: http://forums.adobe.com/message/4556586#4556586
What version of Premiere Elements are you using and on what operating system?
a. export the trimmed video clip from the Timeline to a file for storage on your hard drive
b. you can create subclip copy for the Project Media of the project that you are in, BUT those subclips remain with the project that you are in.
c. depending on what version of Premiere Elements that you are using, you could do a copy/paste between projects with a program such as ClipMate.
Premiere Elements 11 and versions earlier than 8.0/8.0.1 have problems with the ClipMate copy/paste of audio and/or transitions between Premiere Elements projects. Earlier Premiere Elements versions, like 8.0/8.0.1, 9.0/9.0.1, 10 and ClipMate did a good job in copy/paste between Premiere Elements projects.
If you need information on how to export selectively segments from a Timeline, please let us know. The trick to it (besides setting the WorkArea Bar gray tabs) is remembering to put a check mark next to Share WorkArea Bar Only in the export dialog prior to export to file type (where applicable).