OK, it all makes sense. Let me give you a recap.
You seem to have had a server filesystem issue. That explains whey both Akeeba Backup and Admin Tools were broken, you were unable to reinstall over them and had to uninstall before installing again. I'd suggest also checking your free disk space. If it's dangerously close to your quota you'll have many similar issues –not just with our software!– in the near future.
Regarding Admin Tools, since you uninstalled it you lost its configuration. It's advised to use Export before uninstalling and Import right after reinstalling. Since you didn't do that, Admin Tools figured out it's not configured yet and gave you the option to run the Quick Setup Wizard. You accepted that option and that's the "page with questions" you got to. So, it all went according to plan! Our goal was to guide users into configuring Admin Tools easily upon installation.
I'm glad it's all working now! Happy holidays :)
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!