No battery installed magsafe light amber

To the best of my knowledge there is only 1 schematic floating around for this board. It is pre-production and not in any way, shape or form accurate to even the 1.83Ghz Core Duo. There were 17 revisions/versions of the A1181 motherboard. It sounds like you have a short on your motherboard, more than likely caused by liquid damage or a bad solder joint. The components on this board are not really user friendly. They are very small and I would recommend only that highly skilled Technicians with the proper tools work on them. The ICs on this board can only handle 3V - 5V. The average Multimeter uses 9V to run continuity tests and can wipe out one of the ICs if the probes are placed wrong on the board. So you need a continuity tester that only uses 3V or less. Most multitester probes are too large to be able to isolate them to any single point on the board consistently. I have sewing pins soldered on to a set of probes for boards like this. To test the inductors, capacitors, and impedance of the ICs you will need a good LCR meter. One's that can do the job start at about $300. The problem might not be any component. It could be a bad solder joint. The easiest way to find them and see burn marks on components is with a stereo microscope that has at least 45x zoom. You may be able to put a used one of those together with a stand on eBay for about $200. You are more than likely will spend $250 and up for one.

The best advice I can give you is to remove your motherboard and clean it with ISP alcohol 91% or better and a brush. If you had the skills and equipment to do chip level repair on this board you would not be here asking for help. After you have the motherboard out you can inspect for burnt chips and dark spots on the motherboard those are caused by heat and will help you find the problematic parts on the board. Schematics won't help you find problems on a board. Inspection of the board does, both visual and by testing the components.