Is it true that if a son does not bathe and carry the dead body of his father, the father cannot go to Paradise? Is it right that if the son/husband does not carry dead body of the woman or if the woman dies in an unmarried state, she cannot go to Paradise?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

There is absolutely no basis for what you mentioned. It is definitely false, and it contradicts both the Islamic texts and reason.

Allah knows best.