What is the support of a copula, vs support of a marginal?

Continuous marginal distributions have a potentially unbounded support. (does unbounded support mean $$[0,\infty)$$ as much as it could mean $$(-\infty,\infty)$$?)

Is it true then that, on the other hand, a copula:

1. bypasses the estimation of the margins, and
2. the copula has a support in $$[0,1]^2$$ because its margins are uniform?

Why would a range of $$[0,1]^2$$ be desirable over $$[0,1]$$. What is the reason for the exponent $$2$$ with respect to uniform marginals?

Is it true then that, on the other hand, a copula (1) bypasses the estimation of the margins, and (2) the copula has a support in $$[0,1]^2$$ because its margins are uniform?
Why would a range of $$[0,1]^2$$ be desirable over $$[0,1]$$. What is the reason for the exponent $$2$$ with respect to uniform marginals?
Given a bivariate distribution, there are two random variables we are dealing with. Accordingly, the language of mathematics and probability requires two arguments in the joint distribution function as well as in the copula function. It is not that we can choose between $$[0,1]^2$$ and $$[0,1]$$ freely, so the question of desirableness is not really relevant.