I would like to be able to copy geometry of a feature from one layer and paste it on a different editable layer using the
Editor widget on web app builder in ArcGIS Online. However this doesn't seem to have the functionality to do so.
Is it feasible to download the web application, make code changes and upload to AGOL?
No, if you download the application from AGOL the code is minified and hard to read/understand and there is NO ability for you to push your changes back up to AGOL. you would have to host the application on your own web server to be able to use your code changes.