Popping sounds are generally either a sound from a spark, or a failure of an electronic component. They do not always mean that something failed. For example, plugging in electronics that use lots of power will often cause a small spark to form when plugging them in. As an example, my Lenovo laptop power adapter will often make a sparking noise when I plug it into an outlet and it is not a cause for concern.
Does your power adapter create such a noise when using other outlets? If so, then you may not need to worry at all. See this question for some more details on this transient arcing.
But, sometimes noises do indicate failures, for example when a capacitor, MOV or IC explodes. You could also hear a click when a circuit breaker trips, but this would cause a circuit in your house to be no longer be energized. Some power bars also have internal circuit breakers, which can also make a click when they trip.
In your case, I'd first say that a used surge suppressor probably may not provide much protection for your electronics. If you are concerned about surges, you should look into verifying that your house has a good ground, installing a whole-house surge suppressor, using quality surge suppressors where your electronics are installed on both the power lines and data lines such as telephone or CATV. Surge suppressors do age, and can only handle a certain amount of surge during their lifetime. Ensure that new surge suppressors meet the new UL standard 1449 v3 by checking for a holographic sticker that says "Surge Protection Device".
Now to your other questions:
2) This should be safe. Any over currents should be protected by your house's circuit breakers, and a lamp is a pretty "robust" device that wouldn't be seriously damaged by a bad power supply (though the bulb may burn out). That said, do this in an area where if the power strip catches fire, it won't damage anything, and you could quickly pull the plug out of the wall socket if something bad happens. Don't put water on an electrical fire.
3) As I said above, the power bar may not provide significant surge protection, especially now that it may have failed. You may want to dismantle it (while it is unplugged) and see if there are any burnt/charred/damaged components.
This answer is more of a matter of opinion. To be safest, you could replace it with a high quality surge suppressor, but with some investigation, it is likely as safe to use as any other power bar.
When the fridge turns on (some) and when (some) ovens start their capacitors pull a ton of current to start the motor/function needed.
If these are on the same circuit as your TV this is a pretty big issue and the fluctuations could cause damage to your TV circuit board over time.
Even if these aren't on the same circuit your TV is getting a voltage fluctuation from the main circuit. Your TV having intelligence built in probably has its own way of handling these fluctuations - with one turning off some extraneous functions... like HDMI working.
HDMI cables work or don't. If yours is showing a picture then getting a "better" one won't help.
What I would do? First make sure that your TV is not on the same circuit as the fridge and stove. In some places this wouldn't meet code and it certainly isn't good for the electronics.
If they are not on the same circuit then you may need to call the electric company and figure out if they have anything that is causing fluctuations on your line when something high voltage is plugged in. They may fix the issue or offer suggestions - and a lot of times this is free. If the electric company believes it is an issue on your side you will need to troubleshoot why you have this interference - usually a loose connection in the main box.
And lastly you may need to change appliances or tv... Not saying you have to do this but you gave a specific example. Your TV may not handle any kind of voltage fluctuation well. Whenever a major appliance turns on there will often be a little voltage fluctuation. Most electronics are built to handle this. Maybe your TV does not handle this well and your electricity issue is not really a big deal. Or maybe your TV handles this normally and this would happen with all/most TVs.
Question too - do your light flicker or dim during this initially engagement?
Best Answer
Every cable/satellite installer I've ever talked to says "Do Not connect the cable line to a surge suppressor." I'm not sure how exactly these devices are designed, but for some reason they tend to degrade, interrupt, and/or interfere with the signal. The first thing I would try in this situation, is to disconnect the cable from the surge suppressor and connect it using a simple coaxial cable connector.
NOTE: I am not a cable/satellite installer, I'm just some guy on the internet. I will not be held liable and/or responsible for any damage caused by my advice. Proceed at your own risk.