To address the title of your question, to have the ability to adjust rounded corners, start either by drawing a square cornered rectangle or by drawing a path with square corners. If you select a square corned rectangle with the Selection tool, you will see Live Corner widgets. You can drag a widget to adjust the radius on all corners. If the object has been drawn with a tool other than the Rectangle tool, as long as it has corners, you can select it with the Selection tool. Then switch to the Direct Selection tool to see the corner widgets.
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