Are Swing applications really used nowadays? I don\'t find a place where they are used. Is it okay to skip the AWT and Swing package (I learned a bit of the basics though)?
There is no universal question to whether it is ok to skip something or not. It's a matter of priorities. While Swing has a lot to teach you (and it is too heavily infused with AWT), if you're never going to be writing GUIs you may be better off investing in something else.
Swing, while ugly as hell, is still used in many places. A quick search on "java swing" in Indeed or dice would reveal many jobs and industries that require it.