Yes. A sketch:

Taking products with the free G-space EG commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration.

Having done this, there is a natural long exact sequence of homotopy groups

-> pi_* (U) -> pi_*(U_{hG}) -> pi_*(BG) -> ...

and applying this to the pullback diagram you can deduce (from the 5-lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence.

Yes. A sketch:

Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration.

Having done this, there is a natural long exact sequence of homotopy groups

$\to \pi_* (U) \to \pi_*(U_{hG}) \to \pi_*(BG) \to \dots$

and applying this to the pullback diagram you can deduce (from the 5-lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence.