In edit mode.
Edit the polygon.
Select the polygon using one of selection tools.
"Right click the layer -> Export -> Save Selected Features As..."
In "Save Vector Layer As..." window, select a file format and path, then save it.
Now, you can redo the changes on original polygon.
You have a multi-line geometry.
You can use the Merge lines algorithm after you have connected the the two parts, so that they form a continuous line.
You can also split the multi-geometry into two (simple) geometries using the Multipart to single parts algorithm, join them (using the Vertex Tool in the Digitizing Toolbar) to form a continuous line and ...
Another approach if you want an intermediate layer:
select the feature you want to edit and do Ctrl + C to copy feature (or "Edit" > "Copy features")
do Ctrl + Alt + V (or "Edit" > "Paste features As" > "Temporary Scratch Layer...")
edit the scratch layer as you want and right click the layer "Make Permanent" to save (shoter version of right click > "Export"...
Your first error may be from using the vertex tool (all vectors) - it may have tried to modify an underlying layer that you did not intend to edit (for example world_map.gpkg in your error message).
Your second issue is due to the fact that you cannot perform most edits on KML/KMZ files in QGIS. Even though you can see the yellow pencil enabled which ...