In dimensional analysis, how do we go from [T]^1 = [L]^a [M]^b ([L]^c/[T]^2c), knowing that a = 1/2 b = 0 and c = -1/2, to t = (constant) sqrt(h/g)? How do we know there's a constant and where does it come from? Wasn't L supposed to cancel itself out?

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Actually, the equation was [T]^1 is directly proportional to [L]^a [M]^b ([L]^c/[T]^2c), which means that dividing [T]^1 by [L]^a [M]^b ([L]^c/[T]^2c) would give you a non-zero constant. The constant has no dimensions, and does not affect the final dimension of the answer. Unfortunately, this also means that the constant cannot be identified by dimensional analysis. The only way to find the constant is to substitute in each value in the equation and calculate the value of the constant from there.

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Yes. For example, in the equation \[s = ut + \frac{ 1 }{ 2 }a t^{2}\] the constant 1/2 has no dimension and would not be shown in the dimensional equation.
And regarding your second question, L DOES cancel out since \[L ^{a} * L ^{c} = L ^{1/2} * L ^{-1/2} = 1 (dimensionless)\]

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