Is my hastiness in marrying a woman without inquiring about her, then divorcing her, and consequently wronging myself and my son, a sin that warrants punishment?

The Prophet (peace be upon him) guided us to choose a righteous wife, and neglecting this guidance leads to loss. However, there is no sin in rushing into marriage.

It is your duty to spend on your child and to strive to discipline and educate him to the best of your ability. If his mother is not fit for custody, then refer the matter to the Sharia court to have the child removed from her custody. You must do all that you can to care for your child and raise him properly. For anything you are unable to do, there is no blame upon you.