Many water spills damage the keyboard as well as the logic board. Since the power button is a part of the keyboard, I'd suggest that this is why it won't turn on. I actually see this a lot. To test this, there are two pads that you can short together to turn it on without the power button. Refer to this answer to help you find them: Where on the logic board are the power-on pads? Once you've found them, just use a flat screwdriver of similar metal tool to bridge the gap between them. I'd also try disconnecting the battery just in case it is preventing is from booting.
If it was the power supply, you wouldn't get any power at all. From what you're describing it sounds to me like a bad logic board. There was likely a flaky component that was causing your initial Kernel Panics that continued to degrade as time went on.
For me, I'd get rid of it right away. I've seen it tank too many hard drives, and cause so many other issues that I'd never trust it with any of my computers. I'm not saying that is doesn't provide some benefit, but in my experience, that benefit is small, and not worth the cost. Macs in general are pretty good at keeping themselves optimized.
There isn't really a good answer to this, because it really comes down to personal preference. Time Machine is a great program IMO, but it's not prefect. It does get everything backed up, but you have to use the program to restore as well, and I've had issues with this. Especially when the backup drive has issues. And it can be more difficult to go in and retrieve files manually as well. You're locked into the Time Machine system. However, it is extremely easy to set up and maintains itself for the most part. Clones are less simple to setup and maintain, but you can retrieve files easily with Finder, not having to use some program. And if something goes wrong, recovering the backup is usually a simpler process. You're probably fine with Time Machine by itself. For me, having an extra backup is always nice just in case, but that's me. Like I said, it's all about personal preference.
I doubt that the dent on the battery is the culprit here. Typically a battery is fine unless the outer shell is actually punctured (you'd have a fire on your hands in that case). I've mangled more than a few batteries in my time and they always work perfectly fine afterword. I suppose it is possible that the battery is overheating, but my guess is that there was damage to a sensor on the logic board, likely from the fall. I've never heard of anything displaying a overheating problem unless it detects one. Either it's because there really is one, or because one of the sensors is faulty.
That is a common problem with all the classic style iPods. 90% of the time it's just he jack that needs replaced. Over time the contacts inside the jack just wear out. There is a guide here how to do it: iPod Classic Headphone Jack & Hold Switch Replacement The exact part you'll need will depend on exactly which model you have. Some were thinner than others. Part for the "Thin" Model: iPod Classic (Thin) Headphone Jack & Hold Switch Part for the "Thick Model: iPod Classic (Thick) Headphone Jack & Hold Switch
All standard SSD's are 2.5" form factor. The one in the guide is a special Apple drive that you would have a very hard time getting. You can still do this, by replacing the existing 3.5" drive with the 2.5" SSD. All you need it an adapter kit. I'm not sure if iFixit sells them, but I know OWC does. http://eshop.macsales.com/item/NewerTech... There are other ones out there, but be careful, not all of them are designed to work with the iMac's mounting system.
There's nothing between the button on the cable and the aluminum button. The only things I can think or are these: 1) You didn't quite get things lined up correctly. It can be a bit tricky, 2) You could try loosening the screws that hold the button in. Sometimes I see the button doesn't work if they are tightened all the way down.
Try this: Unplug it, pull the battery out. Now, hold the power button down for about ten seconds. Then, plug the magsafe back in and try turning it on. Leave the battery out for now. See if that works. If not, then you either have a bad AC adapter, or there is a hardware problem with the MacBook. You mentioned a flaky DC-in board, it might be just that.
That looks like the cover that goes on the side of the display hinge cover. There's usually one on each side, and it kind of hooks into the hinge. It's purely cosmetic, so I wouldn't worry about it too much. I could be wrong though, haven't taken apart one of these display assemblies in a while.