>> "Internet connection should not be a problem since other machines in the same LAN can connect to the license server, and 'ping wyday.com' from that machine gets response too (btw, is this the proper way to verify internet connection?)."
No, because you're pinging the HTTP side of things (might be intercepted), you're not seeing the result (might be intercepted & modified) and you're not using the same proxies as TurboActivate. The best way to test things is described in the FAQ: https://wyday.com/limelm/help/faq/#internet-error
>> "Do I need to use the new TurboActivate.dat in order to activate the trial? (sorry I don't wanna try it without your confirmation in case I cannot re-produce the error later)."
No. Unverified trials will need a new TurboActivate.dat file for a new trial length. Verified trials use the data that's on the servers (and ignore the trial length in the TurboActivate.dat file).