Mac on local network has generic PC icon
Sounds like you may have enabled file sharing with SMB (Server Message Block, default for Windows, which is why the icon looks like a "generic PC") rather than AFP (Apple Filing Protocol) for the Mac mini. Check to make sure by going to System Preferences > Sharing. Choose File Sharing in the list on the left and click the Options button on the right. 
SMB is useful for sharing files with Windows/Linux-based computers, but if you only have Macs, stick with AFP. Things like permsissions, hidden files, etc. work much better if you use the native filesharing protocol.