One story in the book was about a beautiful young man who could predict the future and left a string of pregnant girlfriends to derail human evolution. The last story in the book was about a rocket ship with a male and female human to seed the next universe as ours died. I found this book on my 8th grade English teacher's bookshelf in 1960.
The first story sounds a lot like "The Golden Man" by Phillip K. Dick. The publication timeframe is good but it does not appear to have been anthologized, only appearing in Dick's own short story collections. I don't recognize the other story at all but maybe someone else will. Here's the covers for the Dick collections. https://www.isfdb.org/cgi-bin/titlecovers.cgi?37578