Hi Jack,
As Sue mentioned above, could you let us know which control panel theme you are using on the site please.
If not the default theme then could you perhaps try that one and see if the problems persist.
Other than that, are you running any add-ons that might be interfering with the Publish page at all?
The fact that the drop-downs work on other pages in the CP (hopefully I read that right?) but not the Publish page is a little strange so let’s get you on the default theme first and ensure you don’t have any add-ons that might be affecting this first if that’s okay?
Thanks,
Mark