I've taken a look at the phpinfo and noticed the server is not running Apache. Instead it's running something called IdeaWebServer (a server I'm not familiar with). It looks like IdeaWebServer isn't close enough in terms of Apache compatibility. I think there are several potential problems with running on a server that is not 100% compatible with Apache. Beyond just getting it to work in the environment, ProcessWire depends on Apache-specific directives in some areas of security, like blocking access to protected files and directories. The same goes for many other CMSs and web applications. Even if we could get things working here by modifying core files, I would be concerned about long term stability and security in this environment, since it isn't fully Apache compatible.
I would switch web hosts if I were you.
Still, if you want to keep at it, it looks to me like the problem lies in the RewriteEngine. As soon as there are GET variables present in the URL, it appears that the RewriteEngine gets disabled. For instance:
This is not correct behavior for the server's RewriteEngine, and is preventing you from even being able to use the admin. Any idea why IdeaWebServer is disabling the RewriteEngine when GET variables are present? I suspect this behavior would interfere with a lot of applications, so am a little surprised by it. But maybe it's something that your web host can advise on. Please let us know what you find.