run fault codes to check for damage done when jumping the battery as a surge will blow the ECM/CPU. If no codes then use the code reader to reset the light.
Have both your battery and alternator inspected. Turn it on, disconnect your negative battery cable, if it dies, you need an alternator, if not, then you may want to test your battery for bad cells.
It would appear so. have a load test done on the battery to check for a failing cell. Check out put of the alternator as it may not be charging the battery as well.