That is true. White doesnโt have to play Rxb4 after Bb4 either.
It doesn't matter whether white takes the bishop or not, in any case black can promote without being taken by white's rook, for example 1...Bb4 2. Ng2 b1=Q, and white can0t take the new queen.
Once white stops the discovered check with Ng2, black can either capture the rook at b7 or try for mate.
I don't think I follow you. Do you mean after 1...Bxf4 2. Ng2? How can black capture the rook? If the pawn promotes, the rook simply takes the new queen (1...Bxf4 2. Ng2 b1=Q 3. Rxb1). And trying for mate without promoting the b2 pawn seems unnecessarily complicated to me (I mean, the position is -5 in favour of black according to the engine, so probably there is some path to mate, but it isn't obvious)
White canโt move to the king to g1 or h1 because the new queen is covering that row.
White moves the king before the pawn promotes: 1...Bxf4 2. Kh1. And again, if you promote, the white rook simply takes the new queen 1...Bxf4 2. Kh1 b1=Q+ 3. Rxb1