I don't have the answer to your questions, but perhaps I have some leads. You may already know all this.
In the 1850 census for Twinsburg, Summit, Ohio there is a William Maxam, age 64, born in Masssachusetts, and wife Martha, age 57, also born in MA.There are children Isaac, 28, born in NY, Joel, 20, born in NY,Ansel, 18, born in OH.There's also a William Maxam, 38, born in NY and wife Hannah and their children elsewhere in Twinsburg.
In the 1880 census the younger William, Joel and Isaac all live next to one another with their wives and families.By that timethe younger William seems to have remarried. Joel is married to a Sarah A. Maxam, age 58, born in NY, father born in CT, mother born in VT.
I don't have a census index for 1860 for Ohio, but it would seem that Joel was probably there in 1860 and you could confirm whether Melvin was listed with him and what his wife's name was.
A couple of things you could try would be to go through the 1850 census for Twinsburg page by page and try to find a Sarah of the right age with parents from the states listed in the 1880 census--assuming that is accurate.Of course the 1860 census might just have one or both of her parents living with her, or a sibling with her, etc.You could try looking for records on microfilm that can be rented from the Family History Library--catalog is online
I didn't find much for Twinsburg itself, but there are Summit County records including a list of marriage licenses for the right time period and some other records that might be helpful.
I could not find Melvin by searching nationally the 1880 census CDs that I have.He may have not been counted, the CDs may be wrong, or maybe he died before then.There are some cemetery lists for Summit County, but I have no idea whether any would help.I guess I'd look at the census for 1860 and 1870 first and see if he was there, and maybe look for a local historical society.
Finally, I looked at the LDS site to see if there might be any credible leads there.According to one file William is a descendant of Samuel Maxim (spelled every possible way) and Elizabeth Gibbs.This is a Mayflower line, recognized by the General Society of Mayflower Descendants, but a man named Raymond Wing has very recently found what looks to me to be very convincing evidence that the Elizabeth Gibbs the Mayflower Society and others have as Samuel's wife may be the wrong one.This is not settled--the GSMD will look into it--but I am personally leaning toward Ray Wing's proposal.I am a descendant of this Samuel and Elizabeth Gibbs and have tried to look into it--it's something that may not be able to be settled, but I believe Ray has certainly cast considerable doubt at the very least on the "official" identity of Elizabeth Gibbs.