Yes you can insert into a map layout when creating it along with other map elements like north arrow, scale etc. I'm new to ArcGIS Pro so not seeing it among other elements.
The projection is from the inserted map. If you want to convey the information on the layout, add a text element specifying what it is.
Once you've added a map to a layout using a map frame, you can change its projection.
To do this, right-click the Map in the Contents Pane and choose properties
Then adjust the coordinate system on the Map Properties window that appears. It is important to know that if you change the map coordinate system in the layout, it is also changed in the map view and vice versa. Map frames on layouts just reflect the map.