Thank you all for your research and help!
I have more information, or potential information about John now than ever! Even if there is a little bit of new confusion added.
I think the Mary Rainford that died in Hunter Street and was buried in the Necropolis is likely to be right! Joshua and Elizabeth were the only relatives in my family I had known to be buried in that cemetery, so there is a chance that her mother may have been to, and the address makes this likely, I think.
I also forgot to point this out before, but there were other Rainford relatives in Liverpool. Elizabeth and Joshua's daughter, Jane's, marriage certificate in 1835 gives Mary Ann Rainford as one of the witnesses. Perhaps John and
Mary may have brought over a sister of Elizabeth, who never married, or a son who had a daughter, but does show there could be quite a few Rainford's.