Can I safely delete items in my "downloaded apps" folder? I want to free up space on my phone and was wondering if I can delete items in my "downloaded apps" folder.
  I know on my computer there is a difference between the downloaded file and the installed application and that once the application is installed the downloaded file can be deleted.  But I wasn't sure about my phone.
Direct answer: Yes you can safely delete the apk you download once it is installed.

Developed answer: Any apk file you download goes to the external storage (either external sdcard or fuse-mounted internal storage that appears as external storage). Installing an apk is just a matter of copying it to internal storage (inside a folder within /data partition). so once installed, the apk becomes redundant, and it is safely to remove the one you downloaded.