Is it permissible to shorten prayers when visiting one's wife and children for a day and a half, and if not, is it obligatory to make up for past prayers that were shortened for years?

As long as your wife and children reside in a particular country, your stay with them is considered a residency that breaks the ruling of travel. Therefore, you are not permitted to shorten the prayers nor to combine the two prayers that share a time, due to the saying of the Prophet (peace be upon him): "Whoever settles in a land should pray the prayer of a resident." Accordingly, it is obligatory for you to make up the prayers you offered as shortened ones. There is no sin upon you as long as you were ignorant of Allah's ruling.