The power supply is indeed located in the top left corner and over the years power supplies have often exhibited buzzing sounds when they are failing. The LED driver board is also in that area though it doesn't have nearly as much power flowing through it so buzzing would be kind of odd. With what you've given us I'd say the power supply is probably a safe bet. With that said, if it fails again I would check the diagnostic LEDs to see where the power flow is failing. Most of the 27" iMacs (before they lost the optical drive) have 4 diagnostic LEDs just to the left of the RAM slots, visible through the grill at the bottom of the Mac. LED 1 indicates trickle power is present, this should always be lit if the Mac is connected to a power source. LED 2 indicates that the logic board has received power from the power supply. This is the one I'd expect to see be either noticeably dimmer or dark compared to the first one if the power supply is flaky. LED 3 indicates the video card is functioning. LED 4 indicates...

Removing the SIM card and/or SIM card tray is not necessary when replacing the battery on an iPhone 4 or 4S. The battery is on the other side of the device from the SIM card, you only need to remove the back panel and disconnect the battery connector before actually replacing the battery. You can find a guide here - iPhone 4S battery replacement guide

Without actually seeing the Mac I can only give you my best guesses based on my experience. Since the Mac seems to run ok while connected to the MagSafe adapter the MagSafe board (DC-in) and logic board are likely ok. To further test this I would try, if one is available, to run the Mac off of a fully charged battery. With this in mind I would expect either the battery or battery connector board to be faulty. If the Mac runs off of a good/charged battery then it's likely just the battery. If it still doesn't run off of or charge the good charged battery then it's likely the battery connector board. The top case (or upper case in iFixit terms) is the most likely component that would need to be replaced in order to fix the keyboard, since the keyboard is built into this component. Hopefully that helps. The big grain of salt with all of this is that there could be damage at the logic board side of these connections. Unless there's physical corrosion visible then replacing the parts may be the only way to tell...

More than likely the top case connection isn't seated properly. I would disassemble it again and make sure to push this connection in until it feels solid. It's a bit awkward to do the first few times so I wouldn't doubt that it might be loose if you haven't removed the top case a few times. Step 12 in this guide shows the cable I speak of. Before re-assembling the Mac I would try to power it on, just to make sure it works before you reinstall all of the screws.

The Mac should still be under warranty considering it hasn't been available a whole year yet. If an Apple store is convenient to your location you could make a Genius Bar appointment at Apple.com/retail to have a Genius pop the bottom case off and blow it out for you. Shouldn't take them more than 5-10 minutes. With that said, if you'd prefer to do it yourself I would refer you to the teardown of this Mac. The fan is readily accessible once you've removed the bottom case and as such should be fairly easy to clean. Working through step four of the aforementioned guide should get you to cleaning the fan. Things to keep in mind if you do tackle the cleaning job yourself... (1) Be sure to disconnect the battery. Even though you don't have to touch the logic board or other sensitive components it is always a good idea to work on a non-powered unit when it's an option. (2) Use quick bursts of compressed air to push debris away from the fan, through the heatsink and out of the Mac. (3) To prevent the fan from...

(image pulled and modified from step 6 of the iFixit MLB replacement guide for this model) I believe the power pads for this model are located at the top of the logic board, just to the right of it's top left mounting point. The two flat silver pads are larger than similar ones in the area. A T8 should work for shorting them. Something to keep in mind is that a damaged or faulty keyboard could prevent a Mac from starting if the power button is always registering a press. If I were troubleshooting this I would first try to short it using the power pads. If that didn't work after a couple of attempts (the flat ones are more difficult to short) I would disconnect the keyboard's flex connection (near the RAM slots) and attempt it again. Good luck!

I can't say with 100% certainty that this is true, however I do have first hand experience repairing and/or refurbishing hundreds (possibly 1,000+) MacBooks. With that in mind, it's my experience that Apple frequently changed mounting locations and made small revisions between MacBook generations. For instance a logic board from an Early-2008 MacBook might physically fit and bolt into a Late-2008 MacBook, but the bluetooth connection on the logic board may be a different type. Alternately the logic board may not fit due to mounting points in the bottom case being slightly different. Thats just one example but it often holds true, especially with the very early MacBooks (yours is a first generation) as Apple made lots of revisions in the first few years. The fan connection, bluetooth connection, speaker connection, optical drive connection, etc all changed pretty early on. Unless you have physical access to several logic boards to test before hand I would recommend you stick with the logic board intended...

So just to clarify, the external display DOES work and display the proper video? If so then your GPU is likely ok. Assuming this these are the steps I would take: __________________________________________________________________ Shine a light into the display while the Mac is powered on. If you can see images/video then your backlight is out. Try adjusting the display brightness using the F1 & F2 keys. If adjusting the backlight brightness has no effect then try a PRAM/NVRAM reset, holding Command+Option+P+R while powering on your Mac, releasing all of the keys after you hear it chime three times. If the issue persists then I would attempt an SMC reset, make sure your Mac is connected to it's power adapter, hold Shift+Command+Option+Power button for about 5 seconds then release all of them together, wait a second then attempt to power on normally. If the issue still persists then you are dealing with a hardware issue. Possibilities include a blown backlight fuse, failed inverter or failed display....

The two power pads are located directly above the trackpad connection on the logic board. The can be seen clearly in step 15 of iFixit's guide Image pulled from the same guide I linked to above and modified to highlight the power pads Bridging these two pads, using a T6 or flat head screwdriver, with power supplied to the Mac should force it to power on. If the Mac still refuses to power on then you might disconnect the keyboard (the flat flex ribbon connection outlined in the red square) as a stuck power button can prevent the power pad method from working. Sometimes just disconnecting/reconnecting this keyboard connection can resolve the issue you describe as well. Good luck!

Funny, I just replied at Stack Exchange as well.. Try this... Power on your Mac while holding the option key on your keyboard. If done correctly, and assuming a firmware password is not set, you should see your internal hard drive show up on the display. If you DO have a firmware password set then you will be prompted to enter the password before moving on. This is just to verify what you should see. This basically tests the option key at the most basic hardware level before the OS is loaded. If you don't see these screens then your key is likely failed at the hardware level. If this is the case then you have to replace the entire top case since the keyboard is built into this component. If the key had popped off you could replace just the key but any sort of functional (non cosmetic) repair requires the keyboard be replaced. The repair for this isn't terribly difficult although it is somewhat involved since the top case is basically the chassis that everything bolts to, meaning you have to tear the...