If f(n) = Θ(g(n)) then what all can one say about f(h(n)) and Θ(g(h(n))
In general, you can prove nothing. You know $f(n) = \Theta(g(n))$. Now assume f (n) = g (n) = n for all n except g (n) = 0 if n ≤ 1000. And h (n) = 1000 for all n. 

So we now have f (h (n)) = 1000 and g (h (n)) = 0. 

The problem is that $O (n)$ and $\Theta (n)$ only work for large n. You are fine if h (n) -> inf, or if you look specifically at f (n) and h (n). 

If the limit of h (n) is infinity, then you overcome the problem that $\Theta(g(n))$ only gives you information for n ≥ N for some possibly large N, because h (n) will eventually be greater than that N.