Dave -
I think there is some general confusion on what BuildAssemblies are. I think before you use the MethodAPI, you should try a bunch of examles in QuickBooks directly.
ItemInventoryAssembly is the actual QuickBooks item. Like "Bicycle M5200". This will appear in QuickBooks under the item list, which shows you the current stock of "Bicycle M5200". Initially, you have a count of 0 "Bicycle M5200" if you just started your business today and just started using QuickBooks today.
ItemGroupOrAssemblyList is a list of all the raw components and other materials that make up the "Bicycle M5200", like wheels, handlebars, spokes, frame, etc.
Now, how do you get some of these "Bicycle M5200"? You need to build them! The way you build them is by creating a BuildAssembly and specifying "Bicycle M5200". The quantity you specify will increase your count of ItemInventoryAssembly in QuickBooks.
If you create a BuildAssembly using the MethodAPI, and specify IsWaitForSyncApproval=True in the Add request, it won't go to QuickBooks until you mark IsWaitForSyncApproval=False.
For your question: "Can Method create API to simulate clicking that SAVE button?", this functionality already exists. Use the MethodAPIActionSendToDesktopV2 webservice call.
HTH
Paul