Not able to run test with dates populated from Datepicker and data from Pop-up screen

I believe that the issue with both of your test examples is that clicking on an object by location is not reliable.  

For instance, in your test log, I see:  

 A click at point (68, 10)

TestComplete will always click at that point, but the box you are trying to click on may not always be at that point, depending on the browser, the screen size, etc.  

It is more reliable to tell TestComplete the object name and have it click there rather than on a particular point.  Then it won't matter where objects are on the screen.