I have to ask: what, exactly, is a "tarheel"?

[theHandpuppet]

And how does that relate to having a goat/ram for a mascot?

-- A curious Buckeye nut

[Clark2008]

1. I don't know why they use a ram

but, one version of the nickname's record talks of a battle in Virginia, where their supporting column retreated, but North Carolina troops fought alone. The victorious troops were asked in a condescending tone by some Virginians, who had retreated, "Any more tar down in the Old North State, boys?" The response came quickly: "No, not a bit; old Jeff's bought it all up." The Virginians asked: "Is that so? What is he going to do with it?" The reply: "He is going to put it on you'ns heels to make you stick better in the next fight."

[Somawas]

3. Because in 1948 Charlie "Choo Choo" Justice was known as

the Battering Ram.

[tsuki]

2. This, I think.

The word Tar Heel (or, Tarheel) is the nickname given to someone who is a native or resident of North Carolina, the Tar Heel State.

The term allegedly comes from the Colonial Era in which tar was one of North Carolina's major products. It came from the workers who walked through the forrest accumulating the tar on their shoes.

A more colorful explanation comes from the Civil War in which North Carolinian soldiers were abandoned by other retreating soldiers to fight alone. Upon returning to the retreaters, the "Tar Heel Boys" told the other soldiers, "Next time we'll put tar on your heels to make them stick," insinuating that they won't be able to flee.
"God bless the Tar Heel boys," General Robert E. Lee exclaimed, on hearing the story.