We've got a new router - which unfortunately has a known bug which stops Microsoft PPTP VPNs working. I spent some time struggling to configure an IPSEC VPN on the router (and gave up when I couldn't find a free VPN client), before trying to get Exchange's RPC over HTTPS connections working.
The server was set up correctly, and all the diagnostics we're reporting that I could make a connection - but Outlook just wasn't connecting to the server. I went through all the documentation I could find, double-checking as I went. Everything was correct - on the laptop and on the server. But still no connections...
Finally I found the answer, hidden in a comment on a discussion forum somewhere in Germany.
As I was self-certifying my server, I needed to install the root certificate as a Trusted Root Certification Authority. I exported the certificate from the server, fired up Internet Properties on the laptop, and installed the file. Finally I fired up Outlook.
(And I sorted out my remote file access issues by installing OpenSSH and WinSCP)