Pretty much comes down to 3 items: 1).Defrost heater 2). Defrost limit switch - Usually a bi-metal switch in series with the heater and shuts it down if it goes over a certain temperature. 3). Defrost timer - The "brains" of the refrigerator. Tells the temperature switch when it is allowed to tell the compressor to run. Takes the temperature switch's control away (every 8, 10, or 12 hours depending on the setup) and tells the defrost heater to work for 20 - 30 minutes (depending on setup). Then gives control back to the temperature switch after the defrost cycle. Slim chance but, still possible, is an ice bullet in the freon line or a burnt wire. I don't know how anyone could know which is the problem child without going to where the refrigerator is and testing it. I have over 25 years experience repairing refrigerators. If you are handy with a multimeter and mechanically inclined, it is a repair you could do yourself. The repair place isn't really steering you wrong, in my opinion.

C2670 is a 18 PF (picofarad) Capacitor rated at 50V (volts) with 5% tolerance in the 402 SMD package. That is very small, few people could solder that back on without a stereo microscope. You will find it here at Digikey. L2451 is a 10 uH (microhenry) Inductor (aka coil or choke), 10uH inductance rating, a current rating of 0.12A (120mA) and 0.36 OHM (360mOHM) in a 603 SMD package. If Digikey has it you will find it among these here. This should be enough information for a brave soul, such as yourself, to find what you are looking for. If you have any more questions on this matter. Please feel free to ask thru comment option. One of us will do our best to answer or guide you. Good Luck!

Install your hard drive. Start your computer and insert the Mac OS X CD. Restart the computer. Then follow the instructions here.

It is highly unlikely that the LCD is the problem. 99.999% of the time if it is faulty you will get something other than a blank white screen. Inspect the cable ends and the sockets they plug into on the boards. Look for corrosion or black burn marks with a magnifying glass of 5x or better. If you find any clean it off. You can use an emery board or sand paper on the male parts. Usually you can get by with a sewing needle on the female socket, a small dremel grinder tool is better (using it with your fingers not with the motorized dremel tool). Check for continuity on LVDS cable, also the left IO cable between it and the logic board with a multimeter. I've seen a burnt connection for the IO cable cause similar problem to this. Since you have replaced the left IO board, with a known good one? I would suspect the problem is a bad connection, bad left IO cable, bad LVDS (video) cable, or the problem resides on the logic board. Confirm your cables and their connections are good. If by small chance the problem...

With out seeing pictures of them, I'd say look at step 24 here for their proper placement.

With you using a cheep APS USP unit and keeping the iMac on the battery, you have protected it. You say you purchased the power supply from iFixit, this one here. This comes with a 1 year warranty. Follow the return procedures given to you by iFixit, they will replace it. There is no need to buy another one.

It has been known to work most of the time, sometimes not. However, putting faster memory sticks in won't buy you better performance. You are limited by the clock speed of the front side bus and memory controller. You are restricted by the design the engineers originally intended the limits to be. Match the existing stick of RAM and you should be fine.

Pull the logic board, pull PRAM battery & heatsink. Put in a small plastic tub. Pour enough 91% or better ISP Alcohol to cover it. Scrub it with a soft toothbrush. Take a small sewing pin and clean the female connectors for the battery cable connector to the board. When you do a SMC reset, the SMC does not preform a pre boot/POST test. Then after the system boots the SMC shakes hands with everything, takes an inventory. If something isn't preforming correctly the SMC tells the fan(s) to remain on high. It then remembers something wasn't working correctly and won't let the system boot, unless you reset it. A good cleaning should fix what is going on here, if it worked correctly when you put it in "storage". Depending on the humidity and temperature levels of where it was stored, the logic board and it's connectors most likely have some corrosion. Don't forget to apply new thermal paste on the CPU & GPU when reseating the heatsink.

Replace your water fill valve or take it apart , remove debris and clean with a product such as CLR. The chances are that the solenoids are good on the fill valve. Most likely it is clogged slightly open with calcium. lime or debris allowing water to trickle through. If you are on a well it is most likely sand or dirt holding the cold water valve open. Clean it out or replace the valve. Then replace the washers to the fill valve(s) with screened washers (where the water hoses attach)to prevent this from happening again. Instructions for removing/replacing valve can be found inthis video. Good Luck!

Reset your SMC. Instructions for doing that are here. What kind of shape is your battery in? What color is the magsafe LED, is it bright or dim? It appears that the NVRAM gets it's power from the main battery on these. What are the last 4 characters of your serial number, to help us confirm the exact model. Resetting the SMC should do it unless there is a problem with the battery or board. Has this been dropped or exposed to liquid?