I'm not sure this is an answer, but it's too long for a comment, and it may be helpful in suggesting some ideas for research.
Bethel College is now Bethel University, and has moved from its founding location in McLemoresville, Carroll County, Tennessee, to McKenzie, Tennessee, which is at the point where Carroll, Henry, and Weakley Counties meet. According to its website, Bethel is associated with the Cumberland Presbyterian Church.
There is a William C. Gardner of the correct age, 22, living in Carroll County, Tennessee, in the 1850 census. He is enumerated with a family that looks like Anderson Siles but Ancestry has transcribed as Anderson Liles. He is listed at the end of the family unit, after what is presumably the last child of Anderson and Dorinda Siles/Liles. He could be a younger brother of Dorinda, a son of Dorinda and a previous husband, or an unrelated lodger. His occupation is not given, but there is a tick mark in the "Attended school within the year" column.
Your Bethel College link gives the catalog for 1851, but maybe that is one year too late; possibly William finished in 1850.
If the William at Bethel and the 1850 census William are yours, there's still the question of where he grew up, which may be Carroll County and may be elsewhere in Tennessee.
Avenues to explore:
- The Bethel College / University link. Perhaps there is a historian at Bethel U that can help you, or maybe the Bethel U library has some old documents in which your ancestor is listed.
- The Presbyterian link. William attended a Presbyterian college, and in the 1870 census his occupation is Pres Minister. Churches keep records. Check records of Presbyterian churches in Carroll County, Tennessee. Since he ended up Presbyterian, there's a strong possibility that he grew up Presbyterian, so perhaps his baptism is recorded somewhere.
- The move from Tennessee to Alabama. Was this in order to take a position at a church?
- The Siles/Liles family. Are they related, or were they just hosting William? Are William and Dorinda related?
- William's middle name is Carroll, and he went to school/lived in Carroll County. Is this just a coincidence, or is he named after the county, or is the county named after his family? Or is he just a foundling who was given the middle name of the county on whose courthouse steps he was found?
- Court records. FamilySearch has probate records and other court records for Carroll County, Tennessee. They may not be digitally searchable, but often these books have a handwritten index at the front that is either roughly alphabetical (i.e. all G on the same page, but not in order) or chronological. Look for someone who might be William's father, and try to find William listed as an heir somewhere.
- Do you know of any siblings that William might have had? When searching for other Gardners, e.g. in court and church records, try "William" as well as any names he may have given his children, since names tend to get passed down in families.
- Are there any Gardner families nearby in the 1840 census with a male child in the 10-15 age range? Can you research them to find out what happened to those families and if they have a relative named William?
- Older copies of the Tennessee Genealogical Society's Ansearchin' News are online. You can search the the FamilySearch catalog to determine which issues you want to look at (e.g. there's a 1966 issue that has cemetery records for a Presbyterian church in Carroll County, Tennessee), and then go to the TNGS site for the actual issue. FamilySearch won't let you view the whole issue because of copyright, but TNGS is the copyright holder, so the TNGS site is the place to read the issues.
Some of these ideas may be dead-ends or may lead you down a path of researching a family only to find it's not yours, but even that is information you didn't have before, so it's not completely wasted effort. Just remember to keep track of what you've researched so you don't retrace your steps.