If we can drag and drop controls in WebPartZone in Browse mode?
Yes, you can . please have view this video if you want to how to drag and drop with webparts
If I can do it with ASP.Net web application? not with AJAX?
I think the user can drag Web Parts controls to edit the layout of the page in design mode, whereas in browse mode only standard Web browsing functionality is available. And you should contains a
WebPartManager control and One or more zones in your pages.
For detailed information, you can refer here