the TV is not grounded is because it has only two metal prongs
This quite normal for Class II Double Isolated equipment which does not require a ground.
Scroll down half way for an explanation of Class II devices with no earth connection requirements.
https://en.wikipedia.org/wiki/Appliance_classes
Since your TV is not grounded, the metal shells of any rear panel connectors on the set, e.g. HDMI inputs, will not necessarily be at ground potential until you connect a cable to a grounded item such as a PC with a normal earthed ATX PSU.
There exists the possibility a potential difference might exist between the shell of an HDMI cable connected to the TV and the mains safety earth in the AC outlet. This might explain the slight shock you experience when holding the HDMI plug in one hand and touching the exposed metal chassis of the PC whilst connecting the video lead.
Although any leakage currents from metal connectors on a Class II device (your TV) should not be high enough to constitute a safety hazard, it can still be disconcerting if you experience a very mild shock.
In these circumstances, it is not good practice to plug the HDMI cable from the TV into your PC whilst the TV is switched on,
or in standby mode.
The only "safe" way to connect a signal cable such as an HDMI lead from a Class II device (in my opinion) is with the TV physically disconnected from the AC mains, either by switching the set off at the wall outlet or unplugging the TV's mains lead from the mains.
Similarly, it is better to make connections between the TV and the computer when the computer is switched off too. This can easily be achieved, if your ATX PSU has a proper mains on/off switch on the back. Better still use the 13A socket's switch if fitted.
This way, with both the TV and the PC disconnected from the mains, neither piece of equipment will be powered on (or in standby) so no potential difference should exist and you won't get a shock when plugging/unplugging the HDMI lead.
Since you're using a standard UK 13A mains extension, there's a pretty good chance your house wiring is fine (if you're living in the UK).
The 17th Edition Wiring Regulations mandate the provision of a mains safety earth connection to all 30A ring main circuits feeding the 13A sockets in your home. The safety earth is often tied to a metal water or gas pipe near your mains distribution unit.
https://www.thenbs.com/knowledge/17th-edition-wiring-regulations
The same is not always true of cheap 13A mains extensions. The earth connection in your 13A extension strip might be faulty. Not common, but it does happen.
I've listened to your video, but apart from "wind noise" from the PSU fan, I can only detect a couple of clicks near the start.
Many ATX PSUs contain a relay which goes click when you apply mains to the PC. This bypasses the input surge components (Varistor) after a few seconds to increase PSU efficiency and a click sound is perfectly normal. You may hear another click from the relay when you unplug the ATX PSU from the mains or flip the rear panel on/off switch.
Any arcing sounds are a sign of trouble. Likely culprits if you switch your PC on/off at the rear every day, is the small mains switch on the back of the ATX PSU.
Despite the provision of switch-on current-surge protection in the form of a Varistor, a momentary current spike of up to 100A flows through the switch contacts each time they close. In the example below, it's 87.6A @ 230V AC.
https://www.tomshardware.com/reviews/how-we-test-psu,4042-7.html
Many ATX PSU switches are rated at only 6A or 10A continuous and arcing between the contacts can degrade the switch rapidly, to the point where replacement may be necessary after prolonged continuous daily use.
If your PSU is only 7 months old, I'd be surprised if the switch has failed so soon, but a friend of mine brought his computer round for repair with an arcing mains switch in the ATX PSU. I discovered he switches his PC on and off at the back of the ATX PSU, four or five times a day. After 5 years, the switch was totally knackered. Blackened contacts, gobs of molten metal inside.
If you are in the habit of flicking the switch on the back of your ATX PSU many times each day, it could be the cause of your arcing problem, if it exists. The switch arcing noise tends to be an intermittent fizzing/crackling sound, which happens most of the time the PSU is powered up and the computer is running.
The only safe solution if the switch is fried is to buy an new PSU and stop using the integral on/off switch too often.
Equally well, something else might be dying in your PSU, or elsewhere in the computer. Or nothing at all. Difficult to say. Try to localize the sound with a cardboard tube used as a stethoscope.