Should a husband help with housework and childcare?

Yes, helping with housework is a sunnah of the Prophet ﷺ and a sign of a husband's good character.
Aisha RA was asked what the Prophet ﷺ did at home, and she answered that he was always helping his family; when the time for prayer came, he would go out to pray. He also sewed his own clothes and mended his own sandals. The Prophet ﷺ said the best of people are those who are best to their families. All the more so when the wife also works, a fair division of tasks is what kindness requires. See Rights of husband & wife.