1. Logfile server does not properly works with such constructions in
index.htm: <A HREF="logs"... , but works with <A HREF="logs/"...
This was a know issue with the directory browsing in the log file server that was fixed in the code for version 6.2.2. However the initial index page that you see when you navigate to the Logfile Server is provided by an HTML file in the WinGate directory, which wasn't updated to suit.
You will need to edit this HTML file and append a trailing backslash to all the links.
C:\Program Files\WinGate\index.htm
Are there any security suites installed on the computer? Have you disabled the Windows Firewall?