The last thing you want is a UI blocking popup like alert() that, at least in browsers like Firefox and IE, effectively lock the entire browser from doing anything, which is a terrible anti-pattern in multi-tabbed browsers. Chrome did it right, at least with the HTTP authentication prompt, to make the UI block on a per-tab basis and have the prompt window not be a top-level-window-manager-managed-block-the-entire-app modal dialog.
I think a case can be made to have functionality that shows a dialog that is modal within the context of the page it is on, not modal for the entire browser. The former would be useful for the cases you outline, the latter is the bane of users everywhere.
(forgive me if I'm misrepresenting the current state of browsers other than Chrome when it comes to modal dialogs like alert(), I don't use them much; I think everyone can get what I'm saying despite that)