I have a Web server which uses port 80, just as Safepipe's public interface does. Can I still map Web traffic on port 80 to a private IP address and use the remote management feature from the public side of Safepipe?

You can still map Web traffic and use remote management. To map all public Web traffic to a private IP address on port 80, you need to configure a 'Static NAT' or 'Port forward' rule in the firewall section on Safepipe. Either rule automatically entails an implicit access rule for traffic to the Web server's private IP address. Once traffic from the public interface is mapped to the private IP address on port 80, you should always begin the URL you use for remote management from Safepipe’s public interface with 'https://...', thus using port 443 instead of port 80.