Is it permissible for my mother, who has owed fasting days since childhood, to feed the poor for all the days first and then fast them, or to fast and then feed the poor, or is it obligatory for her to fast and feed for each day on its own day?

If your mother broke her fast after reaching puberty, and she is now able to make up for it, then she must make up for the missed fasts (qada') and offer expiation (kaffara), which is feeding one poor person for each day, if she delayed the qada' without a valid excuse. There is no objection to offering the feeding before or after the fasting.