Can't remote into Mac mini
Some more info would be helpful here. What app are you using on your iOS devices and Mac to connect to your Mini? What happens when you try to connect—do any errors appear, or does nothing appear to happen?

First off, make sure Screen Sharing is enabled in Sharing Preferences (Remote Management is only for using Apple Remote Desktop, so don't bother with that unless you're actually using that program). Double-check that Allow access is set to either "All Users", or "Only these users" with your user name included in the box.

Now try connecting from another Mac, using the Finder's Connect to Server dialog (under the Go menu, or ⌘K). Under Server Address, type in vnc://IP_OF_YOUR_MINI then hit connect. If you can connect that way, but not using the Finder's Shared panel, then the problem is likely in the Bonjour service that advertises the screen sharing option.

Another thing to check is what version of Apple Remote Desktop Client you have installed. If you have 3.6, there's a bug that can cause timeouts for Screen Sharing/VNC clients. The temporary fix is removing /System/Library/CoreServices/RemoteManagement and re-installing 3.5.3. This worked for me on Lion, I don't know if it will work on Mountain Lion. Make sure to keep a backup copy of the files you delete, in case the install doesn't work.