To Question 1: Just a conjugation. Denoting $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ the components of $\tilde \Phi(x,y,t)$ , we have $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} .$$ So

$$\partial_t\tilde \Phi(x,y,t)= $$$$= \left(\partial_t \Phi(x,t) , \frac{\partial_t\Phi(x + r y,t) - \partial_t\Phi(x,t) }{r} \right)$$ $$= \left(f(\Phi(x,t),t), \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} \right)$$ $$= \left(f(U,t), \frac{f(U+r V,t) - f(U,t)}{r} \right)$$$$=\tilde{f}_r(U,V,t) $$ $$=\tilde{f}_r(\tilde \Phi(x,y,t),t).$$

Question 1: Denoting $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ the components of $\tilde \Phi(x,y,t)$ , we have $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} .$$ So

$$\partial_t\tilde \Phi(x,y,t)= $$$$= \left(\partial_t \Phi(x,t) , \frac{\partial_t\Phi(x + r y,t) - \partial_t\Phi(x,t) }{r} \right)$$ $$= \left(f(\Phi(x,t),t), \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} \right)$$ $$= \left(f(U,t), \frac{f(U+r V,t) - f(U,t)}{r} \right)$$$$=\tilde{f}_r(U,V,t) $$ $$=\tilde{f}_r(\tilde \Phi(x,y,t),t).$$

Question 2: as to $\tilde\mu_t$, the same result of the quoted link apply in particular to the flow $\tilde\Phi$, therefore you still have the Liouville equation $\partial_t {\tilde\mu}_t + \operatorname{div\,} ({\tilde f} {\tilde\mu_t) }= 0$

To Question 1: Just a conjugation. You need to observe that if $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ then $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} $$ so

$$\tilde{f}_r(U,V,t) := \left(f(U,t), \frac{f(U+r V,t) - f(U,t)}{r} \right)$$$$= \left(f(\Phi(x,t),t), \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} \right)$$ 

To Question 1: Just a conjugation. Denoting $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ the components of $\tilde \Phi(x,y,t)$ , we have $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} .$$ So

$$\partial_t\tilde \Phi(x,y,t)= $$$$= \left(\partial_t \Phi(x,t) , \frac{\partial_t\Phi(x + r y,t) - \partial_t\Phi(x,t) }{r} \right)$$ $$= \left(f(\Phi(x,t),t), \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} \right)$$ $$= \left(f(U,t), \frac{f(U+r V,t) - f(U,t)}{r} \right)$$$$=\tilde{f}_r(U,V,t) $$ $$=\tilde{f}_r(\tilde \Phi(x,y,t),t).$$