Found the solution for the problem on my PC. The problem was Outlook. I am running Office 2010. I also discovered that other programs also have this problem/issue, but the pause/freeze is not so long as in PL/SQL Developer.

Today the Microsoft Exchange Server was offline so I just closed Outlook and then suddenly PL/SQL Developer was working as it should. When I started Outlook again the problem was back. I do not know why this was working on my old PC.