Coppermine is not capable to have "sub-admins" or moderators (although this feature has been requested often before), so the answer is "no, sorry - what you're up to is currently not possible with coppermine".
However, with slight modifications you can accomplish something similar to this:
Currently you can set up coppermine to allow users to have a personal gallery (which they in turn can use to create albums within it and set viewing permissions to existing groups that the overall admin has set up). Additionally (or rather, as an alternative), they can set up a password per album and share the password for that particular album with their friends and family (through email or password hint). Of course they can upload files to the albums they created.
You (as coppermine admin) can allow the filetypes your users are allowed to upload - this is not only limited to pics, but documents like PDFs etc. as well. You should of course make sure that your users can not upload executables or other potentially dangerous stuff like PHP or perl scripts (by default, potentially dangerous file types are disabled anyway if your webserver is set up properly).
I suggest setting up a coppermine test install and then start reading the docs that come with it - should give you an idea of what is possible and what not. You're welcome to search the board as well for ideas on how to set up coppermine to fit your needs (there is for example a hack available that will display an actual preview thumbnail of a pdf instead of the generic pdf icon that is usually displayed as a thumbnail for a pdf).