Your first claim is false already for very simple cases.
Take G=H=R (the real line).  Define F(x)=0 if x≤0 and F(x)=exp(−1/x^2) if x>0.
The pullback does not exist in this situation:
the set-theoretical pullback would have to be (−∞,0]⨯(−∞,0]∪{(x,x)|x∈R},
which is clearly not a smooth manifold.