If everything had been working previously in some configuration with the HDMI cable plugged in at both ends then my best guess is that there is a fault in your lamp. Maybe an earth has become detached and the metal base is somehow live. I'm assuming that the HDMI cable was plugged in to something at its other end at the time of the incident.
I believe the outer part of the HDMI plug, shown damaged, is earthed as part of the shielding and all the metalwork of your PC equipment and peripherals should be at a similar earth potential. I don't think the breakers would have tripped if it had been a low voltage event, say one of the PC power supply outputs getting grounded. They are able to withstand that sort of maltreatment and just shut down or current limit. It sounds far more like a mains short of some kind, nothing to do with the actual PC.
My advice is not to use the lamp until you have had it inspected and tested by a competent electrician. It might be extremely dangerous. The lamp fuse might not have blown if the fault was to connect the neutral (rather than the live) to the metal of the base because the fuse is in the live side. Whilst the neutral is nominally at earth potential, in old or badly installed mains wiring it can be quite a way from earth, sufficient to trip an earth leakage breaker if connected to earth via the HDMI cable.
later
Your continuity measurements just about confirm our suspicions. The lamp is faulty, you have a 12 ohm short between live and the metal work and the only thing protecting you is the paint!
Scrap the lamp completely and buy one either with a proper 3 wire earth, or that is marked "double insulated" (plastic body and internals).
The other readings are pretty much what I would expect for a properly grounded system. Where you couldn't get earth continuity try to scrape a bit of the paint off, but in any case those places are not essential.