Flashing ROMs means Installing zipped apps?
Android can only install .apk files (though .apk and .zip are interchangeable)
Yes, it does mean that, but you'll need a custom recovery which has the ability to flash .zip files. Note that such .zip files are not just applications packed in an archive, those .zip files are specifically designed for recovery flashing, and have specific scripts that run when they are flashed
installing custom recovery requires root
As said before, you need root to have recovery so that you could install .zip files. To achieve root access, you'll need a PC.