That does sound like quite a few problems, let's start with the battery and say it's bad, and u ordered a new one so we will wait on the results with that, next u say without the battery it goes through stages of powering on and off and then fully boots, then sometimes will reboot itself or blue screen on u, have u checked to see if this is a heat issue, make sure the fan is not obstructed and is spinning like normal possibly a new fan may be in order, what happened when u tried changing out the MagSafe board as well I'm not sure if that has anything to do with these issues seeing as it will do power on normally and still do all this irregular stuff, and to me it for sure does not sound like a graphics issue, u might check the error codes on the blue screens and fail screen your getting as well