For example, (as per my requirements,) how do I activate in code the button 'Add Data' form the Layer group, and get notified of the end of the operation?
You can use existing ArcGIS Pro commands in your own add-in by using the FrameworkApplication class' GetPlugInWrapper method.
There are a few samples that shows you how to do this:
Hook Pro Commands sample.
Identify Window sample
are you looking for the SDK literature specifically ArcGIS Pro SDK | ArcGIS for Desktop
or how to run the quivalent tool to add a layer in arcgis pro
An overview of the Data Management toolbox—Data Management toolbox | ArcGIS for Desktop
specifically those dealing with layers?
An overview of the Layers and Table Views toolset—Data Management toolbox | ArcGIS for Desktop
I don't want to do the action myself, since the pull-down menu has many options, (and that would be a problem too!)
The thing is that in my UI I have a button with instructions for the user to press on 'Add Data' button, add whatever data s/he pleases then click back my button to continue.
My manager wants that button to automatically do like clicking that button (as I said the pull down menu would be an issue, but for now I have to do what I'm told...)
Retrieving data ...