Is there any installation instructions, and other documentation, for Linux
Ubuntu?

just use the JabRef plugin manager. In JabRef
Plugins -> Manage Plugins
Click "Install" and select the .jar file

Is there a way to push to TeXstudio something like \citet{} or \citep{}
instead of \cite{}?

Cheers

You can set the command in JabRef. Right-click on the "Push selection to TeXstudio" button to open the settings.