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.
Long-term darkening of a fuse on a high-draw circuit like an AC is not unusual. However, darkening as quickly as you have described means the fuse is operating right at its limit for too long.
The first thing I would do is call an electrician and have the fuse replaced with a breaker. At least that way you don't have to touch a hot fuse to replace it. They may also find a problem with the wiring (corrosion or a break) that could explain your problem.
Next is I would buy/borrow a multimeter that has an amp clamp on it. Measure the current used by various parts of your system (blower motor, compressor, condenser fan) as well as the main power feed to find out how much current you are pulling. This may help you identify which component is causing the problem.
Best Answer
There is a short somewhere in the oven where hot is making contact with neutral or ground (the frame). This could be anywhere but the easiest thing to do is start at the plug and work your way up. Check the plug and cable for any signs of damage. If the damage is at the plug or cable then it is something you can fix pretty easily - replacement parts are easy to come by at your local hardware store.
If a quick visual inspection does not show anything you will need to start disassembling the oven and tracing the electrical connections, looking for anything damaged like nicked conductors, missing insulators, etc. If you do not have experience working on electrical components it is best to contact a qualified service technician for your appliance.
Do not use the oven until this is resolved. Do not short out the fuse or try a larger fuse.