Ok, here is how the things have evolved in those days.
The problem is “solved” in the sense it is understood, unfortunately it requires me to trash this router.
Rana works perfectly as it has been set up to use fixed DNS instead of those provided via DHCP.
The router was reset to factory default and I set up a new password with nearly cryptographic strength. One day later, laptop and Q6600 were having the original problem.
The router was inspected and I found the DNS were changed to manual setting.
Restored to dynamic DHCP and reboot. I also cleaned Q6600, which was found to be again positive to hundreds of PUPs (a full AV scan is still pending and hopefully will be carried out this weekend).
I suspect Q6600 might have been setting the router from the inner side. Nonetheless, the fact it guessed the password implies there’s a exploit in the router. Unfortunately, I’m already running the latest firmware so I’m afraid there’s no solution.
In short, I guess I can say the problem is going to be resolved as soon as I get a new router.
Thank you again for your help! You saved me a lot of time and grime! All the best!