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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 is not a Lie groupoid in this situation: the set-theoretical pullback is (−∞,0]⨯(−∞,0]∪{(x,x)|x∈R}, which is clearly not a smooth manifold.