Even though I've now reset the password again, I keep getting the "Sign in Required" box. This is when I'm trying to install an app.
MIght be due to third party software > Mac App Store: Sign in sheet does not appear, or does not accept typed text
If you have an app called Little Snitch instaled, it can cause the problems you have described.
If you have anti virus software installed, disable that.
Retrieving data ...