The blinking folder could be just the MacOS which needs to be reinstalled or a defecte harddrive. But shutting down is off course the biggest problem. Is there any moisture damage ?

If you are sure the battery was full of charged then it must be the motherboard. You have to have good understanding and knowledge of electronics to repair motherboards so if you don't have that you have to let someone else repair the motherboard for you but keep the repair price below €100 - €150 ($110 - $165) because this T100 costs around €300 euro ($332)

It could be a charger problem (charger or connector in the T100) so the battery isn't charged and maybe drained at this moment. Do you know if the battery was fully charged ?