What is the ruling on a husband having sexual intercourse with his wife in her family's home after the marriage contract has been concluded, the dowry paid, and the marriage announced? And is the husband sinful if he does not perform ablution and pray two rak'ahs before intercourse, as mentioned in some hadiths?

There is no harm in a husband being intimate with his wife in her parents' home, as long as privacy is secured away from sights and sounds. As for not performing the two rak'ahs of prayer before consummating the marriage, there is no sin in that, as it is not obligatory, and no elevated hadith (definition) has been authenticated concerning it. Rather, some scholars deemed it desirable because it was reported from some Companions.