If parents reconcile after divorce, and then things go bad again, will the father owe back child support/alimony from when they were reconciled?

It is possible that you will technically owe support and allimony, but if all the parties were living together under the same roof, you can apply to the court to have that waived as not being applicable.  you should have proof as to the dates on which everyone was back together and when child support/alimony should start again.