Yes, when a creature is regenerated, it is removed from combat. If the effect of a resolving spell or ability regenerates a permanent, it creates a replacement effect that protects the permanent the next time it would be destroyed this turn.

Does regenerate work with combat damage?

Unless the creature is dealt lethal damage or destroyed, the regeneration effect will not be applied so the creature will still be attacking and still deal/receive combat damage as normal.

Does a regenerated creature stay tapped?

A creature that has been dealt lethal damage and been regenerated gets tapped afterwards. However, creatures who are already tapped still can be regenerated, but remain tapped.