Regarding ‘Hajr-e-Aswad’ & ‘Riyadh Al-Jannah’

I need your opinion on two aspects of some of the general muslim beliefs:

1. It is a widely held Muslim belief that Hajr-e-Aswad (the black stone) is a stone from paradise and bears some white marks at its face. It is further believed that these white marks or spots will turn black as the judgment draws near.
2. I had the opportunity of visiting Masjid-e-Nabvi in Madinah. I found out that adjacent to prophet (sws)'s Roza inside the mosque, an area has been identified separately as a piece of land believed to be from paradise. It is referred to as Riaz-al-jannat. Before making the visit many of my friends and relatives advised me to offer Nafals in that area, with the belief that this area specifically holds a very special significance for acceptance of prayers.
Do these beliefs hold any significance and acceptability as per your understanding of the Islamic history.
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