Well, the marriage of Horace and Alice took place on 12 Mar 1885, Polk County, IA, and Alice was 19, so it appears that she was born about 1866ish, that is according to the record mentioned in my answer above.
Henry "W." McElderry and Rachel Williams were married on 26 October 1856 in Tuscarawas County, Ohio.("Ohio Marriages, 1800-1958," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XDVY-Q65 : accessed 27 May 2016), Henry W. Mc Elderry and Rachel Williams, 26 Oct 1856; citing Tuscarawas,Ohio, reference ; FHL microfilm 317,307.) Both apparently died in Harrison Co., Iowa, in 1898.
In 1870, Greenfield, Warren, Iowa, a Joseph Beck, a 30 year old farmer b. England, was living with the "McEldey" family: Albert McEldey, 22 b. OH; Margaret McEldey, 63 b. OH; Margaret McEldey, 30 b. OH; and James McEldey, 27 b. OH. The Thomas Beck family, Thomas 29 b. ENG, was enumerated immediately above the McEldeys.("United States Census, 1870," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MDKS-LB7 : accessed 27 May 2016), Margart Mceldey in household of Albert Mceldey, Iowa, United States; citing p. 8, family 45, NARA microfilm publication M593 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 545,922.) Note that there are no Beck children with this family, and Joseph is a farmer. This would seem to show that this Joseph Beck and Margaret were not married in 1870.
An 1880 US Census in Sevastopol Twp., Polk, Iowa, shows a Joseph Beck, a 35 year old coal miner, living with wife Margaret, 35 b. OH, and 4 Beck children born in OH, including Alice, 13, and Albert, 8. So this couple was married by 1867. This appears to the the correct family.
In the 1885 Iowa State Census for Des Moines, Polk, Iowa, Joseph J. (indexed as G.) Beck, 45 b. ENG, was a Supt of Mine. Living with him were wife Margaret, 44 b. OH; Horace G. McEldery, a 24 year old bookkeeper b. OH; Alice McEldery, 18 b. OH; Albert E. Beck, 13 b. OH; and two younger Becks. So here, Alice and Horace are living with her parents.("Iowa State Census, 1885," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:HWFG-G6Z : accessed 27 May 2016), Joseph G Beck, Des Moines, Polk, Iowa; citing p. 13, 1885, State Historical Society, Des Moines; FHL microfilm 1,020,173.)
In 1900, Joseph Beck, 58 b. ENG Nov 1841, a coal mine operator, was living in Des Moines, Polk, Iowa, with his wife Margaret, 61 b. Ohio Aug 1838. They have been married for 35 years, and Margaret has had 8 children, 4 of whom are living. Son Albert, 29 b. Iowa Aug 1870, is living with them.("United States Census, 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M9LH-Y8Y : accessed 27 May 2016), Joseph J Beck, Des Moines Township, Precinct 1 Des Moines city Ward 4, Polk, Iowa, United States; citing sheet 14B, family 319, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,240,454.) So this would mean that this Joseph and Margaret were married about 1865. But son Albert is shown as too young to be the same Albert as in the 1880 and 1885 censuses?
So, based on the differences between the 1870 census in WARREN County, Iowa, and the 1880, 1885 and 1900 censuses in POLK County, Iowa, it is possible that there were two different Joseph Becks married to two different Margarets, but I haven't found any time when the two families appear on the same census. I think you need to build this out one record at a time and examine it critically to see if it supports the conclusion that Alice and Horace were cousins.