Suppose g is an odd function and let h = f ∘ g. is h always an odd function?

No we can not say that h will always an odd function. Because we don't know about the function f.

As we know Multiplication of two odd function is even function. And Multiplication of one even and one odd function is an odd function.

So if f is an odd function then we can say h is a multiplication of two odd function and multiplication of f and g will be an even function. So h will be an even function.

Similarly if f is an even function then h is a multiplication of one even and one odd function and multiplication of f and g will be an odd function.