Wow, that's quite a list of broken things.

Without knowing much about your system -
which linux OS, and which version number (or how old) it's hard
to prescribe a good answer.

However- I would try the following to see if it gets you closer to getting
the system behaving as it did before.

Can't install a program via GNOME app installer? No worries - perhaps you have
Synaptic? If not, sorry for assuming.

Can't shut down or reboot without supplying a password? Well, I say just make
sure
you have saved any documents you're working on first, and then unplug the
machine.
It's extreme yes, and you may lose some data yes, but it will possibly
tell your computer to do some checking of the HD upon turning back on again.

Once computer boots into operating system, use the GNOME updater tool (again
this
varies from distro to distro - but you get the idea - make sure you have all
the latest software from the distro.