I haven't really used plasmoids (desktop widgets) before, but want to try them out. However, it seems that when I add one, they are not visible. Sometimes one appears for a moment, but I don't know why (mouse hovering?) and also I can interact with them (tried with the analog clock: right-click works, clicking brings up month view widget) if I find where they are. So it seems they are there, but just hidden. Should I press a shortcut to make them visible (temporarily)?

That is odd. Plasmoids usually have well-defined backgrounds or at least should always be visible (unless they are blending in with the background too well).
Try changing the desktop theme under System Settings -> Workspace Theme -> Desktop Theme to see if the problem is with some specific theme or is more general.
You may also try changing the desktop layout by right-clicking on the background -> Desktop Settings -> Layout (options should include Folder View which displays the contents of ~/Desktop in addition to plasmoids and Desktop which only shows plasmoids).

Screenshots of what you are seeing will be helpful in understanding what may be going on.

After a restart, the plasmoids appeared. Not sure what the problem was, but it seems to have solved itself.