I am having the exact same issue as described above (new modules and pages disappearing and reappearing), only I am using DNN version 04.06.02, but I can recall it happening prior to upgrading well.

This issue exists with the parent portal as well as child portals.  So far, the only way I can temporarily resolve this is to restart the application whenever I notice it happeneing. However, I have several dozen child portals that users are managing themselves, and they do not have access to restart the application when their new pages and modules are disappearing randomly, so that is not a real solution.

Does anyone have any suggestions on how to resolve this?

Shouldn't design mode just hide the content  in the module, but still display the container and the page?  In my case, I'm not in Design mode (and I toggled it on and off to make sure)... and I still have the entire page disappearing.  I've tested this on several different portals with the same results. 

Here is an example scenario:  I have a portal (parent or child, doesn't matter, I get the same results) with let's say 3 pages (Page1, Page2, Page3).  I add a new page called TestPage.  TestPage has View permissions set for all users and is set to show in the menu.  I click on Page1 and the link for TestPage disappears from the navigation menu.  I click on Page2 and the TestPage link reappears.  Click on Page3 and TestPage is still there.  Click on Page1 again and TestPage disappears.  If I'm on TestPage and I refresh the page and everything looks ok; refresh the page again and the page disappears and bounces me to the home page as if the page never existed.  If I look for the page under Admin > Pages, TestPage is not listed... if I refresh Admin > Pages a few times then TestPage shows up again.  This continues on and on until I restart the application.  No more issues with TestPage exist, but when I try to add a new page, then that page will exhibit the same random behaviour, until I restart the application.

The same goes for a module, if I try to add content, sometimes it updates, sometimes it doesn't.  If I delete a module from the page, it randomly shows up on the page again.  Sometimes it's there, sometimes it's not.  Also, the page keeps randomly switching itself from Edit mode to View mode when a page is refreshed or a post back happens, so it sometimes takes me 3 or 4 tries of updating a module's content before it'll stick.

This is a big issue for me as I have several dozen portals that this is affecting and a lot of annoyed and confused users.  I haven't been able to find anything in the forums that would suggest a fix.  I tried searching for something in Gemini, but got an application error when using the search tool, so I gave up on that.  Any ideas would be welcome!