Remove and clean the board using 95%+ isopropyl first, then test it. The parts you mention don't look blown, they are stained by some sort of liquid. Once cleaned, if it is still not working, you will need to download a schematic and board view for your MacBook (board number begins with 820 and will be somewhere on your board, use Google to find the files). Once you have these, you can then use a multimeter to measure voltage rails.
The board screws can be removed using a T5. Let us know how the cleaning goes and we can take it from there.