Will it hurt the chances of my stepson gaining an immigrant visa?

No, it will not hurt his chances. The step son has to be under the age of 21 and your marriage to his father has to have occurred prior to him turning 18 if you are to petition for him.

Yes. You can sponsor both and each case will be considered on its own merits.

You must file a separate I-130 for each. Your husband's overstay should not be a problem for the step child's application.

This is a common occurrence and it is not all that problematic. You will be required to file an I-130 for your husband and your step son. In the category of spouses, unmarried children (under 21) and parents of US citizens, there are no derivative benefits available with these petitions. Meaning that your step son will not derive any benefit from the visa petition you file on behalf of your spouse. I would suggest that you file the I-130 for your step son and the adjustment of status package for your spouse together - but keep in mind that they do not go to the same filing location. Each application package will be processed and it is very likely that your spouse's applications will be completed prior to your step-son's Form I-130 being sent for processing at the US Consulate overseas. Often, we do see that I-130's filed for children overseas will not be processed until the biological parent's applications are completed in the US so this could delay the process a bit. But my advice would be to file both.