I can answer some of your questions. I am still trying to piece the Hays family together, but I can tell you what I know. SOme of this information may be incorrect. William Hays (1810-1892) married Surrelda(1819-1900), I did not know her last name until now. He did not marry Rebecca Baxter. His nephew William did. There children are as follows:
James (1847-1882) Albert(1855-1934) + Mary Johnson (married 4/17/71) John (1849-1935) Elizabeth (1854-1937) + ? Peterson Emeline ( 1846-1928) + Joseph W. Laird (1831 -1887 ) Elizabeth E. Hays
I believe William's father's name was William as well, and Martha was his mother. She was born in Ireland.
If you want additional information please contact me via my email. I hope this helps!