How should motorists react when highway construction shifts lanes across the median to occupy a lane or two on the opposite side?

It varies between states, and whatever the DOT of that state determines is a safe speed. You get out to the Dakotas, they'll probably allow 65 once the crossover is made, whereas a state like Indiana...

You should turn from the right-hand lane closest to the curb/shoulder, into the right-hand lane (closest to the curb/shoulder) of the intersecting roadway.

Slower moving traffic.

If there is only one turning lane, then you should be in the left hand lane. If there are multiple turning lanes, going to the same street, then you stay in your lane during the turn.

ONLY when there's a wide shoulder or a parking lot, and the highway is straight and not near a curve straight ahead.