I'm sorry to hear that you're having trouble with Showkase.
I hope my notes below help.
First of all, please let me try to reassure you that installing and using Showkase should be a quick, easy and error-free process.
I have free web hosting accounts with three different web hosts that I use for occasional personal use and Showkase runs fine out-of-the-box on all three (AwardSpace, ByetHost and x10Hosting).
I won't go out of my way to recommend any of the web hosts that I use as they are free accounts with limited features and no priority support. Having said that, Showakse runs on all of them without issue.
The System Requirements noted here are really all that should be necessary.
Your problem certainly seems to stem from GoDaddy's migration of your site.
Normally, after migrating a site, you'd need to run Setup ('System -> Setup' from the Showkase admin) to let Showkase know the new location of the site.
Also, if a site fails, then I'd also recommend trying 'System -> Repairs -> Repair' (to rebuild the page records).
Finally, if the above two suggestions fail, I'd suggest installing a fresh Showkase site (in a separate directory so that it does not interfere with your existing site) to see if this works (and if it does, then you could import the pages from your existing site via 'Site -> Import' to avoid having to recreate them all from scratch).
Incidentally, it is still worth making a complete backup of your dysfunctional site just now so that you can import pages from it into a new site at some point in the future.
However, the File Test failures that you posted (from the Server Compatibility Test) suggest that nothing I've mentioned above is likely to work and that the problem is an incorrectly configured server.
The first File Test is simply trying to create a new directory on the server.
If GoDaddy's support cannot figure out why a PHP script is unable to create a new directory, then I'm really not sure what to suggest. That's pretty much as basic as a file management task can be.
The fact that you mentioned "GoDaddy no longer supports html coding" is especially worrying!
I think the clue is in the word 'Plesk' (in the File Test paths).
As far as I am aware, GoDaddy uses Plesk for its Windows servers and I've seen issues with GoDaddy/Plesk/Windows in the past.
I wonder if GoDaddy would be willing to move you over to a Linux server (being that their migration has broken your site and they can't fix it)?
I have a hunch (it's a strong hunch, not just a long-shot) that running Showkase on a GoDaddy Linux server will work fine.
Showkase should work fine on a Windows server (I've seen it myself) but there definitely seems to be something amiss with yours and I'm pretty sure that, if you stay with GoDaddy, the way forward would be to migrate (again) to a Linux server.
Please let me know how you get on and if I can be of any further help.
Thank you.