"Casper" is the Germanic form of the Persian name, "Jasper" (one of the "Three Wise Men" of Christian mythology).

I may be wrong, but Paul and Daniel appear to be Ursula's children from a former marriage, whom Joseph has adopted.  Or they may be children from a former marriage of his, but given Ursula's age, more likely they are hers.

— born in Ashford Twp., Fond du Lac Co., WI:
6.  Joseph STROUP, b. 1853/4
7.  Bridget STROUP, b. 1856/7; d. by 1870

1.  Columbiana County, Ohio, Naturalization Records, 1811-1862.  Columbiana County Ohio Genealogy Society (online at the Columbiana County OHGenWeb site):

Record #	Surname	First Name	Origin	Document	Year
2416	STROUP	Casper	Wurtemburgh	Intention	1834
2417	STROUP	Casper	Wurtemburgh	Naturalization	1848

The earlier record must be our subject because his first cousin, Casper STRAUB, didn't come to America until 1843.  The later record is likely his cousin.

2.  Anon.  1966.  Marriage Index:  Ohio, 1789-1850.  Liahona Research, Orem, UT (Broderbund CD-400):

Stroup, Joseph

Spidle, Usila

29 Aug 1843

Columbiana Co.

3.  1850 Census Index/Images (online at Ancestry.com; Image #14 of 15):  Town[ship] of Ashford, District 9, Fond Du Lac Co., WI, p. 417B, 2684/2704, enumerated 20 Nov 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):¤•

Jasper Stroup	30	M		Farmer	1200	Gm
Ursula    "	36	F				"
Paul      "	10	M				Ohio
Daniel    "	8	M				"
Joseph    "	6	M				"
Jacob     "	4	M				"

Are Paul and Daniel Joseph's children from a prior marriage?  Or are they Ursula's?  Given Ursula's age, my bet is that they are her children from a prior marriage and that, being toddler and infant when Joseph married her, he adopted them.  But where's Andrew?  Listed two households from Joseph's first cousin and brother-in-law, Casper STROUP. 

3.  Wisconsin Census, 1820-90 (online at Ancestry.com):

1855 State Census	Straub, Casper	WI	Fond Du Lac Co.	Ashford
1855 State Census	Straub, Joseph	WI	Fond Du Lac Co.	Ashford

Joseph and his first cousin, Casper, are the only two STRAUB/STROUPs in Fond Du Lac County. 

4.  1860 Census Index/Images (online at Genealogy.com, Image #36 of 44):  Ashford P.O., Ashford [Twp.], Fond Du Lac Co., WI, Roll 1407 (Book 1), p. 224, PN 36, 265/255, enumerated 20 Jun 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen):¤•

Uscula Stroup	45	M		W Farmer	3000	975	Wertemburg
Paul     "	18	M					Ohio
Daniel   "	16	M					"
Jacob    "	14	M					"
Andrew   "	12	M					"
Joseph   "	6	M					Wisconsin
Bridget  "	3	F					"

Where was Andrew in 1850?  Did the enumerator miss him?

5.  1870 Census Index/Images (online at Genealogy.com, Image #21 of 46):  Armstrong Corners P.O., Town[ship] of Ashford, Fond Du Lac Co., WI, Roll 1713 (Book 1), p. 31A, PN 21, 144/141, enumerated 15 Jul 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):¤•

Stroup Arsula/Ursula	58	F	W	Keeping House	12500	3000	Wetemburg
"   Daniel	25	M	W	At Home			Ohio
"   Jacob	24	M	W				"
"   Andrew	23	M	W				"
"   Joseph	15	M	W				Wisc
Norber Henry	25	M	W	Farm Hand			Ohio

As in 1850, the enumerator called townships, "towns."

6.  1880 Census Every-Name-Index/Images (online at Ancestry.com, Image #28 of 47):  Ashford [Twp.], Fond du Lac Co., WI, Roll T9_1425, p. 32D, PN 28, SD 1, ED 31, enumerated 19/21 Jun 1880, official enumeration date 1 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

24

25

26

206

209

Straub Andrew

W

M

27

Husband

/

Farmer

Ohio

Germany

Germany

______ Anna

W

F

20

Wife

Keeping House

Wisconsin

Do

Do

______ Ursula

W

F

68

Mother

Retired

Germany

Do

Do

Folm Joseph

W

M

21

Hired Man

/

Farm Labor

Wisconsin

Do

Do

7.  "ash_stmartins" [St. Martin's Catholic Church Cemetery, Ashford, Ashford Twp., Fond du Lac Co., WI] (online at the Fond du Lac County WIGenWeb site):

Name	Date of Birth	Date of Death	Note
Straub Ursula	26 Feb 1814	15 Jul 1889	gattin/v Jos.; geb. BIEGELMAIER