*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  


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