Asymptotic Notation sanity check for f(x) and g(x)

Question

For each of the 2 pairs of functions I need to figure out the following:

1. g(n) = Θ(h(n))?
2. g(n) = O(h(n))?
3. g(n) = Ω(h(n))?
4. g(n) = o(h(n))?
5. g(n) = ω(h(n))?

Pair one g(n) = (64)^(n/4), h(n) = 256^(n/8)

Pair two g(n) = n¹⁰²⁴, h(n) 2^{(lg(n)*lg(n))}, lg is base 2

For pair one, I have the limit as n->inf for g(n)/h(n) = inf. Therefore:

1. g(n) = Θ(h(n)). No, given g(n) = O(h(n)) is not true
2. g(n) = O(h(n)). No, if g(n)/h(n) headed towards 0 and not infinity
3. g(n) = Ω(h(n)). Yes, given g(n) completely overtakes h(n) in speed as n increases
4. g(n) = o(h(n)). No, given g(n) should never be below h(n) for some n0 > 0.
5. g(n) = ω(h(n)). I think yes, given g(n) = Ω(h(n)) is true but g(n) = O(h(n)) is not. Which was the same logic for o(h(n)) in pair one.

So for pair two, I have the limit as n->inf for g(n)/h(n) = 0. Therefore:

1. g(n) = Θ(h(n)). No, given g(n) = Ω(h(n)) is not true
2. g(n) = O(h(n)). Yes, given g(n) completely overtaken by h(n) in speed
3. g(n) = Ω(h(n)). No, if g(n)/h(n) headed towards infinity and not 0
4. g(n) = o(h(n)). Yes, given g(n) = O(h(n)) is true but g(n) = Θ(h(n)) is not
5. g(n) = ω(h(n)). I'm thinking no, same reasoning as g(n) = Ω(h(n)) being no

Made edits based on user plop comments.