Remember to stick to defintions. You live and die by them in mathematics. Recall that imaginary numbers have their own definition. They are the set of all real multiplies of i, the sqrt(-1). So, no, ∞ is not a member of that set either.
The appearance of ∞ in interval notation is a shorthand way of saying include all reals above the real number beforehand (a, ∞) or all reals below the one afterward (-∞, b).
The definition of a closed interval is that it contains its endpoint. As these intervals have no endpoint to contain, they are regarded as open (on the relevant side)(the example you gave [3, ∞) is a closed on left-side and open on the right-side).