The ".gz" and ".xz" are extensions automatically appended to a file name when compressed with the gzip and the xz stream compressors, it's not a good idea to swap them. The "file" command utility, if you have any Linux distribution at hand, can tell you if a file was compressed with one or the other.
To install a distribution to the internal HD you probably need to carve some space on it to create and format a partition, install on it and investigate on how to load the kernel and what it needs to start "init" from the PS4 side.
The "start-psxitarch.sh" it's probably a Bash script used to boot/install Arch, even without looking at it I'm pretty sure, with some effort, can me modified to boot any distribution...