My HR manager has posted a job on Careers and is forwarding me applications to review (as I'm the hiring manager), but I cannot see those applications using my Stack Exchange account (I get a 404 error). Is it possible for her to add me as a member of the organisation with privileges to view applications and applicant's profiles?
No, this is not possible. Currently the best way to do this is with the Share Profiles link in the candidate manager. However, account organizations are in our pipeline.