I have no explanation why child No. 5 is Luther (a male) in 1870 and Lucy (a female) in 1880.  It could be enumerator error or the infant may have had  ambiguous genitalia, leading to a decision to change the child's gender as it developed (it happens).

1a.  Jordan R. Dodd, ed.  1993.  Virginia Marriages:  Early to 1850.  Liahona Research, Orem, UT (Broderbund CD-229):

Cupp, Joshua

Karicofe, Catharine

12 Aug 1841

Augusta Co.

1b.  LDS.  Family Search: Internet Genealogy Service:  IGI - International Genealogical Index (online at FamilySearch.org).

Joshua CUPP Spouse: Elizabeth J. BLAGG Marriage: 30 Dec 1864, Augusta, Virginia Source: patron submission

2.  1850 Census Index/Images (online at Ancestry.com; Image #59 of 116):  Rockingham Co., VA, p. 30A, 384/387, enumerated 31 Jul 1850 (extracted by Diana Gale Matthiesen):

Joshua Cup	34	M	Farmer	2500	Va
Catharine "	33	F			"
Henry F   "	4	M			"
John W    "	3/12	M			"
Michael Evick	39	M	Shoemaker		"
William   "	6	M			"
Margaret J Cup	2	F			"

Margaret is our subjects' granddaughter.

3a.  AIS (Accelerated Index System) 1860 Census Index (online at Ancestry.com):

1860

Dupp, Joshua

VA

Rockingham Co.

Harrisonburg P.O.

p. 456

3b.  1860 Census Images (online at Ancestry.com; Image #120-121 of 536):  Harrisonburg P.O., Rockingham Co., VA, pp. 456-457, PN 120-121, 890/866, enumerated 10 Jul 1860 (extracted by Diana Gale Matthiesen):

Joshua Cupp	49	M		Farmer	3500	700	Va
Catherine "	42	F					"
[next page]
Henry Cupp	14	M					Va
John W	10	M					"
Rebecca	5	F					"
Margaret	13	F					"

4.  1870 Census Images (online at Ancestry.com; Image #238 of 612):  Harrisonburg P.O., Franklin Twp., Rockingham Co., VA, p. 118B, PN 4, 23/23, enumerated 22 Jun 1870 (extracted by Diana Gale Matthiesen):

Cupp Joshua	57	M	W	Farmer	3580	1000	Va
____ Elizabeth	34	F	W				Va
____ Eliza C	4	F	W				Va
____ Luther P	1	M	W				Va
____ John W	20	M	W	Works on Farm			Va
____ Rebecca F	15	F	W				Va
____ Margaret	21	F	W				Va

5a.  1880 United States Census and National Index (LDS Family History Resource File: CD-ROM Library):

Census Place: Ashby, Rockingham, Virginia
Source: FHL Film 1255388; NARA Film T9-1388; Pg 447A
	Relation	Sex	Marr	Race	Age	Birthplace	Occupation	Fa	Mo
Josh CUPPS	Self	M	M	W	66	VA	Farmer	VA	VA
Elizabeth S. CUPPS	Wife	F	M	W	46	VA		VA	VA
Eliza C. CUPPS	Dau	F	S	W	15	VA		VA	VA
Lucy P. CUPPS	Dau	F	S	W	11	VA		VA	VA
Hugh S. CUPPS	Son	M	S	W	9	VA		VA	VA
M...S Josh CUPPS	Son	M	S	W	5	VA		VA	VA

5b.  1880 Census Images (online at Ancestry.com; Image #5 of 46):  Ashby Dist., Rockingham Co., VA, p. 447A, PN 5, SD 4, ED 78, enumerated 3 Jun 1880 (extracted by Diana Gale Matthiesen):

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).

1

2

3

4

5

6

8

9

10

13

21

24

25

26

30

31

Cupps Josh

W

M

66

F.

/

Farmer

Va

Va

Va

_____ Elizabeth S

W

F

44

Wife

/

Va

Va

Va

_____ Eliza C

W

F

15

Daughter

/

Va

Va

Va

_____ Lucy P

W

F

11

Daughter

/

/

Va

Va

Va

_____ Hugh S

W

M

9

Son

/

/

Va

Va

Va

_____ M????? Josh?

W

M

5

Son

/

Va

Va

Va

Lucy should be Luther.  Is this census-taker error, or was Lucy of ambiguous gender as an infant?

6.  Rockingham County, Virginia Births, 1853-96 (online at Ancestry.com):

Last Name	First Name	Birth Date	Race	Sex	Father	Mother	Notes	Pg
Cupp	Elizabeth	15 Dec 1865	W	F	Joshua	Elizabeth		158
Cupp	Luther	Aug 1868	W	M	Joshua	Elizabeth		197

7.  Anon.  1875.  Historical Atlas of Allen County, Ohio. H.H. Hardesty & Co. Publ., Chicago, IL (transcribed by Diane Mancini for the Ohio Biographies Project online at RootsWeb.com):

Bath Township... HENRY F. CUPP's parents, Joshua and Catharine, were both born in Virginia.  His mother died in 1863.  His father still resides in Rockingham County, Virginia, where Henry F. was born, October 15, 1845, and where his wife, also, was  born, September 21, 1847.  They were married in 1868, and have three children--Nettie L., John W.H. and Joshua F.  Mrs. Cupp is Mary C., daughter of John and Sarah Weaver, who came to Allen County from Virginia in the year 1840.  Their children are Mary C. (Mrs. Cupp), Hannah E., Sarah V. (Mrs. Henry Irvine), Samantha L., Olive B., Frank, Louisa, Liddie, Melissa and J. William.  Mr. Cupp has four brothers--John W., R.F., E.C., L.P. and H.S.