I don't think its quite that simple.
This set you describe forms a vector space, where the vectors are functions f,g,etc...
it doesn't make sense to multiply vectors. meaning [f[g+h]](x) means add the vectors g + h, multiply by vector f and evaulate it at a point x from the field .
instead for part 1. h(x)[(f+g)(x)] = h(x)[f(x)+g(x)] = h(x)f(x) + h(x)g(x)
Since by the vector space axioms (f+g)(x) = f(x) + g(x). so the above holds.
The same problem with #2,
It might seem as though its a small problem but the brackets and paranthesis can make the statement mean something completely different.