Hi again! I've used this weekend to make sure one more time whether I was right saying that the GavickPro team made the said programming mistake. So now I have to tell that I was wrong saying this. There were many factors which could cause the dropdown items of the Mainmenu didn't work. And I checked all of those as I think. The result is the following. I found a component which I installed on my website and this component blocked normal performance of the dropdown Mainmenu items. The component is WEBO Site SpeedUp (free version). It includes some plugins.
As for my rolling-back to J! 1.5.23, this step was the first one: I just restored my website by means of its old version from October 31, 2011. This website version was without WEBO Site SpeedUp but my first idea about the cause of my problem was not the said component. I thought that updating from J! 1.5.23 to J! 1.5.24 caused the problem. Then I thought that it was the GavickPro team which didn't prepare the Postnote Quickstart template for further Joomla! updating. Hence, as I thought, the GavickPro team made a programming mistake. This was perhaps the easiest way of thinking but now it's clear it was not right.
WEBO Site SpeedUp component causes some risks as I read earlier. And so I just faced the risks. After removing the said component, my website, including the dropdown Mainmenu, works fine (so far). And this is on J! 1.5.24. So now I see that there is no ground to blame the GavickPro team. To be frank, I like very much what the team does. And I wish all the GavickPro team members every success in the future. This is what I wanted to say closing the issue.