1) I am using WiX Installer to install my game on the client, I have my start menu for the game but how do I add one for my WyUpdate.exe and how can I make it use a custom icon.
Are you asking how to make a shortcut in wyBuild or in WiX? In wyBuild you just right click a file, then click "Create shortcut" then drag it to the "Start menu" folder in wyBuild. You can't set custom shortcut icons currently in wyBuild.
2) Can I actually make a custom icon for the actual wyUpdate.exe?
Yes, but you have to recompile wyUpdate from source code. See: How to make a custom version of wyUpdate.