Will we be burdened with sin for refusing our male cousin and his mother to stay with us, knowing that we are four young men and it is impermissible for her to stay with us?

Scholars differ on the ruling concerning hospitality, with the majority holding that it is not obligatory at all. Some, however, view it as obligatory, citing the hadith: "If you stay with a people, and they offer you what is proper for a guest, then accept it. If they do not, then take from them the right of the guest." Therefore, there is no harm in refusing your cousin's hospitality, especially if it entails mixing with those who are not mahram to you.