I have downloaded a free map onto my computer and I can view it there but I can't move it into "my collection". If I can't move it into my collection I can't move it into "my device". I have an etrex20 and a PC with Windows 10.
Not quite sure if I understand... But "collections" are not for maps, they are for your user data - tracks, waypoints, routes. Maps work independently and are selected by the dropdown menu.
Garmin MapInstall is used to install maps, try connecting your GPS and running it directly. You should be able to choose the map you want in MapInstall, then choose whether to send all or part of it.