Is the hadith that mentions that Abu Hurayrah's mother (may Allah be pleased with him) cursed the Prophet (peace be upon him) and then embraced Islam authentic, and did she come with him from Yemen while she was a polytheist?

The story of Abu Hurayrah's mother embracing Islam is narrated in Sahih (definition) Muslim. In summary, Abu Hurayrah used to invite his mother, who was a polytheist, to Islam. One day, he invited her, and she spoke ill of the Messenger of Allah (peace and blessings be upon him). So, he went to the Messenger of Allah, weeping, and asked him to pray for his mother's guidance. The Messenger of Allah (peace and blessings be upon him) prayed for her, saying: "O Allah, guide Abu Hurayrah's mother." Abu Hurayrah returned rejoicing. When he reached the door, he heard his mother's voice as she was bathing. Then she came out and declared her testimony of faith. Abu Hurayrah returned to the Messenger of Allah (peace and blessings be upon him), weeping with joy, to inform him of her conversion. Then he asked the Messenger of Allah to pray that he and his mother be beloved to Allah's believing servants, and that they (the believers) be beloved to them. So the Messenger of Allah (peace and blessings be upon him) prayed, saying: "O Allah, make these two servants of Yours—meaning Abu Hurayrah and his mother—beloved to Your believing servants, and make the believers beloved to them."

As for her coming to Medina with her son, there is no explicit text stating this, but it is more likely that she came with him due to his constant companionship with her.