Yes, it's like this with both Firefox and IE. I don't think we can get around it.
Interestingly, if you have two people on the record, you CAN enter details for the 2nd person, and the pull-down menus display the list of countries and date fields correctly. So that tells you it's not a browser issue but an incompetent-IT issue.
You can enter contact info, and they used the e-mail address i entered to send a confirmation e-mail so that part works.