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Your first claim is false already for very simple cases. Take $G=H=\mathbb R$ (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 $(−\infty,0]⨯(−\infty,0]\cup\{(x,x)|x\in \mathbb R\}$, which is clearly not a smooth manifold.